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Algebraic, extrcmal & metric combinatorics, M-M. DEZA, P FRANKL & I.G. ROSENBERG (eds) Whitehead groups of finite groups, ROBERT OLIVER Linear algebraic monoids, MOHAN S. PUTCHA Number theory and dynamical systems, M. DODSON & J. VICKERS (eds) Operator algebras and applications, 1, D. EVANS & M. TAKESAKI (eds) Operator algebras and applications, 2, D. EVANS & M.'i'AKESAKI (eds) Analysis at Urbana, I, E. BERKSON, T. PECK, & J. UHL (eds) Analysis at Urbana, II, E. BERKSON, T. PECK, & J. UIIL (eds) Advances in homotopy theory, S. SALAMON, B STEER & W. SUTHERLAND (eds) Geometric aspects of Banach spaces, E M. PEINADOR and A. RODES (eds) Surveys in combinatorics 1989, J. SIEMONS (ed) The geometry of jet bundles, D J. SAUNDERS The ergodic theory of discrete groups, PETER J. NICIHOLLS Introduction to uniform spaces, I.M. JAMES Ilomological questions in local algebra, JAN R ST'ROOKER Cohen-Macaulay modules over Cohcn-Macaulay rings, Y. YOSHINO Continuous and discrete modules, S.H. MOIIAMED & B J. MULLER Helices and vector bundles, A N. RUDAKOV et at Solitons nonlinear evolution equations & inverse scattering, M. ABLOWI'IZ & P. CI.ARKSON Geometry of low-dimensional manifolds 1, S. DONALDSON & C.B. THOMAS (eds) Geometry of low-dimensional manifolds 2, S. DONALDSON & C B. THOMAS (eds) Oligomorphic permutation groups, P. CAMERON L-functions and arithmetic, J. COATES & M.J. TAYLOR (eds) Number theory and cryptography, J. LOXTON (ed) Classification theories of polarized varieties, TAKAO I.1JJITA Twistors in mathematics and physics, 'I'.N. BAILEY & R J. BASTON (eds) Analytic pro-p groups, J.D. DIXON, M.P.F DU SAUTOY, A. MANN & D. SEGAL Geometry of Banach spaces, P F.X. MU)LLER & W. SCIIACIIERMAYER (eds) Groups St Andrews 1989 volume 1, C.M CAMPBELL & E F. ROBERTSON (eds) Groups St Andrews 1989 volume 2, C M CAMPBELL. & E F. ROBERTSON (eds) Lectures on block theory, BURKHARI) KULSHAMMER Harmonic analysis and representation theory for groups acting on homogeneous trees, A. FIGA-T'ALAMANCA & C. NEBBIA Topics in varieties of group representations, S.M. VOVSI Quasi-symmetric designs, M.S. SitRIKANDE & S.S. SANE Groups, combinatoncs & geometry, M.W. LIEBECK & J. SAXL (eds) Surveys in combinatorics, 1991, A.D. KEEDWELL (ed) Stochastic analysis, M.T. BARLOW & N.H. BINGHAM (eds) Representations of algebras, H. TACHIKAWA & S. BRENNER (eds) Boolean function complexity, M.S. PATERSON (ed) Manifolds with singularities and the Adams-Novikov spectral sequence, B. BOTVINNIK Squares, A R. RAJWADE Algebraic varieties, GEORGE R. KEMPF Discrete groups and geometry, W.J. HARVEY & C. MACI.ACIILAN (eds) Lectures on mechanics, J.E. MARSDEN Adams memorial symposium on algebraic topology 1, N. RAY & G. WALKER (eds) Adams memorial symposium on algebraic topology 2, N RAY & G. WALKER (eds) Applications of categories in computer science, M.P FOURMAN, P.T. JOHNSTONE, & A.M. PITT'S (eds) Lower K- and L-theory, A RANICKI Complex projective geometry, G. ELLINGSRUD, C. PFSKINE, G. SACCHIERO & S.A STROMME (eds) Lectures on crgodic theory and Pcsin theory on compact manifolds, M POLLICOTT Geometric group theory I, G A. NIBLO & M A ROLLER (eds) Geometric group theory II, G.A. NIBLO & M A. ROLLER (cds)
Shintani zeta functions, A YUKIE Arithmetical functions, W. SCHWARZ & J SPILKER Representations of solvable groups, O. MANZ & T.R. WOLF Complexity: knots, colourings and counting, D J.A. WELSH Surveys in combinatorics, 1993, K. WALKER (ed) Locally presentable and accessible categories, J. ADAMEK & J. ROSICKY
Polynomial invariants of finite groups, DJ BENSON Finite geometry and combinatorics, F DE CLERCK el at Symplectic geometry, D. SALAMON (cd) Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI W. METZLER & A J. SIERADSKI (eds) The algebraic characterization of geometric 4-manifolds, J.A. HILLMAN
London Mathematical Society Lecture Note Series. 184
Arithmetical Functions
An Introduction to Elementary and Analytic Properties of Arithmetic Functions and to some of their Almost-Periodic Properties
Wolfgang Schwarz Johann Wolfgang Goethe-Universitt t, Frankfurt am Main Jurgen Spilker
Freiburg im Breisgau
CAMBRIDGE UNIVERSITY PRESS
Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1994 First published 1994 Printed in Great Britain at the University Press, Cambridge
British Library cataloguing in publication data available Library of Congress cataloguing in publication data available
ISBN 0 521 42725 8
To OUR Wives DORIS and HELGA
Contents xi . . . . . . . . . . . . . . . . . . . . . .. . preface xv . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . .. . xvii Notation Tools from Number Theory . . . . . . . . . . . . 1 Chapter I 2 I.I. Partial Summation . . . . . . . . . . . . . . . . . . . . . 1.2. Arithmetical Functions, Convolution, Mdbius Inversion Formula 4 15 1.3. Periodic Functions, Even Functions, Ramanujan Sums 19 1.4. The Turin-Kubillus Inequality . . . . . . . . . . . . . . I.S. Generating Functions, Dirichlet Series . . . . . . . 25 . 31 1.6. Some Results on Prime Numbers . . . . . . . . . . . . 1.7. Characters, L-Functions, Primes in Arithmetic Progressions 3S . . . . . . . . . . . . . . . 39 1.8. Exercises . . . . . . . . . 43 . . . . . . . . . . . . . . . . . . . . Photographs .
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Chapter II
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Mean-Value Theorems and Multiplicative Functions, I
11.1. Motivation
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11.2. Elementary Mean-Value Theorems (Wlntner, Axer) . . 11.3. Estimates for Sums over Multiplicative Functions (Rankin's Trick)
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11.4. Wirsing's Mean-Value Theorem for Sums over Non-Negative 65 Multiplicative Functions . . . . . . . . . . . . . . II.S. The Theorem of G. Halasz on Mean-Values of ComplexValued Multiplicative Functions . . . . . . . . . 76 11.6. The Theorem of Daboussi and Delange on the Fourier-Coefficients of Multiplicative Functions 78 . . . . 11.7. Application of the Daboussi-Delange Theorem to a Problem of Uniform Distribution . . . . . . . . . . . . . . 81 82 11.8. The Theorem of Saffari and Daboussi, I. . . . . . . . . 11.9. Daboussi's Elementary Proof of the Prime Number Theorem 85 11.10. Mohan Nair's Elementary Method in Prime Number Theory 91 .
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Contents 11.11. Exercises
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Related Arithmetical Functions . . . . . 111.1. Introduction, Motivation
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III.S. On a Theorem of L. Lucht . . . . . . . . 111.6. The Theorem of Saffari and Daboussi, II 111.7. Application to Almost-Periodic Functions 111.8. Exercises
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111.2. Main Results . . . . . . . . . . 111.3. Lemmata, Proof of Theorem 2.3 111.4. Applications . . . . . . . . . . .
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Chapter IV
Uniformly Almost-Periodic Arithmetical Functions . . . . . . . . . . . . . . . . . . . . IV.1. Even and Periodic Arithmetical Functions . . . . . IV.2. Simple Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.3. Limit Distributions IV.4. Gelfand's Theory: Maximal Ideal Spaces . . . . IV.4.A. The maximal ideal space 0B of ,$U IV.4.B. The maximal ideal space 0., of Bu IV.S. Application of Tietze's Extension Theorem . . . IV.6. Integration of Uniformly Almost-Even Functions .
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Ramanujan Expansions of Functions in 8" . Chapter V V.1. Introduction . . . . . . . . . . . . . . . . . . . . V.2. Equivalence of Theorems 1.2, 1.3, 1.4, 1.S . . . . . .
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Almost-Periodic and Almost-Even Arithmetical . . . . . . . .. . . . . . . . ... . . . . Functions VI.1. Besicovich Norm, Spaces of Almost Periodic Functions VI.2. Some Properties of Spaces of q-Almost-Periodic Functions
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Chapter VI
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185
186 197
Contents
VI.3. Parseval's Equation . . . . . . . . . . . . . . . . . . . VI.4. A Second Proof for Parseval's Formula . . . . . . . VI.S. An Approximation for Functions in S1 . . . . . . . VI.6. Limit Distributions of Arithmetical Functions . . VI.7. Arithmetical Applications . . . . . . . . . . . . VI.7. A. Mean-Values, Limit Distributions . . . . VI.7.B. Applications to Power-Series with Multiplicative Coefficients . . . . . . . . . . . . . . . . . VI.7.C. Power Series Bounded on the Negative Real Axis VI.8. A 2 q - Criterion . . . . . . . . . . . . . . . . . . . . . . VI.9. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Photographs .
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Chapter VII The Theorems of Elliott and Daboussi 233 VII.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 234 VII.2. Multiplicative Functions with Mean-Value M(f) * 0, Satis.
fying
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VII.3. Criteria for Multiplicative Functions to Belong to 21
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VII.4. Criteria for Multiplicative Functions to Belong to 2q
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VII.S. Multiplicative Functions in Aq with Mean-Value M(f) $ 0 VII.6. Multiplicative Functions in ,4" with Non-Void Spectrum VII.7. Exercises . . . . . . . . . . . . . . . . . . . . . .
257
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Chapter VIII Ramanujan Expansions . . . . . . . . . . . . . . VIII.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . VIII.2. Wintner's Criterion . . . . . . . . . . . . . . . . . . VIII.3. Mean-Value Formulae for Multiplicative Functions VIII.4. Formulae for Ramanujan Coefficients . . . . . . . . VIII.S. Pointwise Convergence of Ramanujan Expansions .
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VIII.6. Still Another Proof for Parseval's Equation VIII.7. Additive Functions . . . . . . . . . . . . . . VIII. 8. Exercises
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Contents
Chapter IX Mean-Value Theorems and Multiplicative Functions, II IX.1. On Wirsing's Mean-Value Theorem . . . . . . . . . IX.2. Proof of Theorem 1.4 . . . . . . . . . . . . . . . . . IX.3. The Mean-Value Theorem of Gabor Halasz . . . . IX.4. Proof of Proposition 3.3 . . . . . . . . . . . . . . . . . .
IX.S. Exercises Photographs Appendix
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A.I. The Stone-Weierstrass Theorem, Tietze's Theorem A.2. Elementary Theory of Hilbert Space . . . . . . . . . . . . . . . A.3. Integration . . . . . . . . . . . . A.4. Tauberian Theorems (Hardy-Littlewood-Karamata, Landau-Ikehara)
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A.S. The Continuity Theorem for Characteristic Functions A.6. Gelfand's Theory of Commutative Banach Algebras A.7. Infinite Products . . . . . . . . . . . . . . . . . .
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A. 8. The Large Sieve A.9. Dirichlet Series
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Bibliography
Author Index
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Subject Index . . . Photographs . . . Acknowledgements
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367
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Preface
This book is an attempt to provide an Introduction to some parts, more or less important, of a subfield of elementary and analytic number theory, namely the field of arithmetical functions. There have been countless contributions to this field, but a general theory of arithmetical
functions does not exist, as yet. Interesting questions which may be asked for arithmetical functions or "sequences" are, for example,
the size of such functions, (2) the behaviour in the mean, (3) the local behaviour, (4) algebraic properties of spaces of arithmetical functions, (S) the approximability of arithmetical functions by "simpler" ones. (1)
In this book, we are mainly concerned with questions (2), (4) and (5).
In particular, we aim to present elementary and analytic results on mean-values of arithmetical functions, and to provide some insight into the connections between arithmetical functions, elements of functional analysis, and the theory of almost-periodic functions.
Of course, standard methods of number theory, such as the use of convolution arguments, TAuBERIAN Theorems, or detailed, skilful
estimates of sums over arithmetical functions are used and given in our book. But we also concentrate on some of the methods which are not so common in analytic number theory, and which, perhaps for -
xi -
Preface
precisely this reason, have not been refined as have the above. In respect of applications and connections with functional analysis, our book may be considered, in part, as providing special, detailed examples of well-developed theories.
We do not presuppose much background in these theories; In fact, only
the rudiments of functional analysis are required, and we are ever hopeful that mathematicians better acquainted with this theory may provide yet further applications. In the Interest of speedy reference, some of the material is gathered in an appendix to the book.
Our book is not intended to be a textbook. In spite of this, some of the chapters could be used In courses on analytic number theory. Both authors quite independently, have led courses on arithmetical functions, and the present text is - In part - an extended version of these courses, in particular of lectures on arithmetical functions given in Frankfurt
am Main and In Freiburg Im Breisgau in the 1992 summer term to third- and fourth-year students.
Our book presupposes some knowledge of the theory of complex functions, some fundamental Ideas and basic theorems of functional analysis and - on two or three occasions - a little knowledge of the theory of integration. Some acquaintance with elementary number theory would be helpful, and [sometimes] a good deal of patience in performing long and troublesome calculations Is demanded.
An attentive reader will notice that certain techniques are used again
and again, and this may be Interpreted as a hint to develop these techniques independently into a universally applicable scheme. We have
attempted to do this for one particular case in Chapter III, where a general theorem on "related" arithmetical functions is presented with some applications. The underlying idea is to replace multiplicative arithmetical functions by "related", simpler ones. Thus, It is often possible to reduce proofs of complicated theorems to simpler special cases.
- xii -
Preface
The main topics of the book are the following: - a study of elementary properties of arithmetical functions centered on the concept of convolution of arithmetical functions; - a study of mean-values of arithmetical functions, In part by simple, in part by more complicated, elementary methods, and by analytic methods;
- the study of spaces of arithmetical functions defined as the completion of the spaces of even, respectively periodic, functions; - the characterization of arithmetically Interesting functions (in particular multiplicative functions) In these spaces: we discuss Important theorems by P. D. T. A. ELLIOTT, H. DELANGE and H. DABOUSSI. The more general theorems of K.-H. INDLEKOFER 11980] will not be proved In this book, and INDLEKOFER's "New method In Probabilistic Number Theory" (1993) will not be dealt with.
The idea of presenting a book on arithmetical functions grew out of a series of papers presented by the authors, beginning in 1971. Our aim was to replace some number-theoretical techniques, as far as possible [for us] by "soft" techniques that are more common in mathematics.
The papers mentioned and this book itself are an attempt to draw together number theory and some aspects of main-stream mathematics.
We have tried to write the book for third- and fourth-year mathematics
students rather than for specialists In number theory, and we have tried to produce a book which is more or less self-contained. Exercises
of varying degrees of difficulty are given at the end of most of the chapters. These are Intended to provide material leading to greater insight Into some of the methods used In number theory by applying these to more or less special problems. "Pictures" of arithmetical functions give some impression of the behaviour of [well-known] arithmetical functions. Hopefully, visualization of arithmetical functions will be helpful for some readers; mathematics is abstract, but concrete, two-dimensional geometry can illustrate abstract - xiii -
Preface
Ideas of arithmetical functions. Of course, those diagrams that Illustrate inequalities are not intended to be proofs for these inequalities; proofs could be provided by any first-year student, by means of the TAYLOR
formula, for example, or using similar techniques. However, in the authors' opinion, a diagram is both striking and convincing, while an exact proof is often tedious. The relevant literature on the topics treated in the book is enormous, and we thus had to omit many important and interesting results from the bibliography. However, an extensive list of references is given, for example, in ELLIOTT's books.
There are many books which deal with arithmetical functions, some of which we list below, although we feel that there are distinct differences between these and our own book. K. CHANDRASEKHARAN [1970]; his Arithmetical Functions deal with analytic aspects of prime number theory, making use of the properties of the RIEMANN zeta-function and of estimates of exponential sums, P. J. MC C ARTHY's Arithmetical Functions 119861, and R. SIVARAMAKRISHNAN, Classical Theory of Arithmetical Functions, 119891.
Texts covering topics similar to ours seem to be those by P. D. T. A. ELLIOTT (1979, 1980a], J. KUBILIUS [1964], and J. KNOPFMACHER
[1975]. Many interesting aspects of a theory of arithmetical functions may be found in the books by G. H. HARDY & E. M. WRIGHT 119S6 ], L. K. HuA 119821, and T. APOSTOL 119761.
Preface
Acknowledgements
The authors are solely responsible for any errors still remaining. However, they are grateful to Rainer TscHIERSCH for generous assistance with some proof-reading. The manuscript was written on an ATARI 1040 ST Computer, using the word processing system SIGNUM2 designed by F. SCHMERBECK, Appli-
cation Systems, Heidelberg, which in the authors' opinion seems to be suitable for the preparation of mathematical texts.
The diagrams, intended to give some indication of the behaviour of arithmetical functions, were produced by the first author, using the PASCAL-SC system (A PASCAL Extension for Scientific Computation)
created by U. KULISCH and his group at the university of Karlsruhe (version for the ATARI ST, A. TEUBNER Verlag); this said author alone is responsible for programming errors or inaccuracies.
The cartoons at the beginning of each chapter were designed by the artist ULRIKE Dt1KER from Stegen, and we are grateful for her kind assistance.
For help with photographs and permission for publication we are grateful to many mathematicians and to some Institutions (for example Miss VORHAUER (Ulm), The Mathematisches Forschungsinstitut Oberwolfach, The Librarian of the Trinity College, Cambridge and many others). Their help Is acknowledged on page 367.
Finally we wish to thank the staff of Cambridge University Press, in particular DAVID TRANAH and ROGER ASTLEY, and an unknown lector
for their help and patience during the preparation of this book. Wolfgang Schwarz & JUrgen Spilker,
August 1993
Notation a) Standard Notation for Some Sets
N={
1,
2, ...
}, the set of positive Integers,
No = IN U {0) z = ( ..., -2, -1, 0, 1, 2, ... }, the set of Integers, Q = { b ; a, b E 7L, b * 0 }, the set of rational numbers,
the set of real numbers, C the set of complex numbers, real [imaginary] part of z C, Re(z), lm(z) B(a,r) = { z c C; Iz - a( < r the set of prime numbers, 9' [the letter p [in general] denotes a prime] u(,v4) is the number of elements of the [finite] set s4, is the [additive] group of integers mod m, 7L/m7L x (7L/m7L) is the [multiplicative] group of residue-classes mod m, prime IR
E
to M.
b) Divisibility, Factorization
gcd(a,b): greatest common divisor of [the integers] a and b; often also written as (a,b); lcm[a.b]: lowest common multiple of [the Integers] a and b,
din: d is a divisor of n, d 4 n: d does not divide n, pklin: pk is the exact power of the prime p, dividing n: pkIn, but pk+i n, pv,(n) n= gives the prime factor decomposition of n according to pin
the fundamental theorem of elementary number theory, P(n) denotes [sometimes] the maximal prime divisor of n. c) Some Notation for Intervals and Functions on IR
[13] denotes the greatest integer s B (where 0 is real), (B) = B - [B] is the fractional part of the real number B, - xvli -
Notation B0(13) = 13 - [13] - 2 is the first BERNOULLI-polynomial [sometimes also
denoted by 4(13) - we avoid this notation], [x,13] closed interval (X c R; a s x s ]x,13 [ open interval (x e IR; a < x < Q }, x li x = li(2) + f { log u)_' du is the Integral logarithm, L° = 0.577 2...
2
EULER's constant, (2ni'x n),
e,,: n '- exp r(x) = f o ti-1
e-t dt, the Gamma-Function, exponential function and logarithm function, O( ... ), o( ... ) are LANDAU'S symbols; f = O(g) is sometimes also written as f
inner product in 82, '(X) is the vector-space of continuous complex-valued functions defined on the [topological] space X, f ° : continuous image of f E ,$u Du A u in e(A ) L°( AD), 1?°( A,4) under :
the GELFAND transform. h) Some Special Series
S1(f) _ Z P 1 (f(p) P
S2'(f) _ >p,If(P)Is5/4
1 ),
S2(f) = E P
p-1
P-1
I f(p) - 112,
If(p) - 112, S2 q, (f) _ XP,If(P)I>5/4 P-1
.
If(p)Iq,
S3,q(f) = IP Ek>2 p_k .If(Pk)Iq,
0q = If multiplicative, S1(f), S2'(f), S2,q" M , and S3,q M are convergent}.
Chapter I Tools from Number Theory
Abstract. This preparatory chapter forms the basis of our presentation of arithmetical functions. Such techniques as EULER's summation formula and partial summation are Introduced, as is the notion of convolution. Examples of standard arithmetical functions are provided; some properties of RAMANUJAN sums are Introduced, and MbBlus Inversion formulae are proved. The TuRAN-KuBILius Inequality Is discussed, prior to its application in Chapter II, VI, and IX. Finally many results from prime number theory (including some results on characters and the prime number theorem in arithmetic progressions) are presented without proofs.
Tools from Number Theory
2
I.I. PARTIAL SUMMATION
Assume for some given complex-valued function a: n H a(n), defined on the set No of non-negative integers, that some knowledge concerning the sum nsx a(n) is available; then the problem of obtaining information about the sum >
nsx
where g: [0,co[ -4 C is a sufficiently smooth function (think of g(n) = n°C or g(n) = log n, for example) can often easily be solved using partial summation. The following version of this technique is taken from PRACHAR [19S7].
Theorem 1.1 (Partial summation). Assume that a sequence an of complex
numbers, and a sequence an of real numbers, satisfying al < a ... , A n
2
'
gcd(a,d)=1
gcd(b,t) =1
I a sd
bst
gcd(b,t)=1
gcd(a,d)=1 0 =
,
if
r
msr
e(d m)
e(t m)
d $ t, or a+b * 0 mod d,
la-d
lbsd
gcd(a,d)=1
1
= p(d) otherwise.
gcd(b,d)=t
a+b= 0 mod d The reason for the last equality-sign is that for every a there is exactly one b, satisfying a + b = 0 mod d. 11
In Chapter IV we shall need some special values of cr(n). If the index r is a prime power pk, then, as is easily verified, -
pk
(3.4)
cpJn)=
- pk-1 0
,
p k-1
if p In, if pk-111n, if pk-1 t n.
Tools from Number Theory
18
Figures 1.6 and 1.7 illustrate the periodic behaviour of RAMANUJAN 30 and sums rather instructively. The functions c r with index r r = 210, resprectively, are plotted in the range 1 s n s 299. 10
so
loo
lso
200
300
Figure 1.6: RAMANUJAN sum c30 In the range 1 s n s 299 10
too
200
300
Figure 1.7 RAMANLIJAN sum c210 In the range 1 s n s 299
Other examples of r-even functions are 1, if gcd(n,r) = d, gd: n H 1
0, otherwise.
The functions gd, where dir, as well as the RAMANWAN sums cd,
1.3. Periodic Functions, Even Functions, Ramanujan Sums
where
19
dlr, form a basis of the C-vector-space of r-even functions
(this space is of dimension t(r)). This is obvious for the functions gd, and for the RAMANUJAN sums the assertion easily follows from the orthogonality relations. a The KRONECKER-LEGENDRE symbol P is equal to zero if pla; otherwise, if p]' a, it is equal to 1 or -1 If a is a quadratic residue
[resp. non-residue] modulo the prime p. (p) is a completely multiplicative, p-periodic function (considered as a function of the "nominator" a). For a thorough investigation of the LEGENDRE symbol as a function of its "denominator" p, see, for example, H. HASSE [1964]. This function a
Generally, given a character X of the group
(
7L/m7L )
x
of residue-
classes prime to m, in other words, given a group-homomorphism X : ( Z/mZ )x --) ( C,
.
),
IX(n)I = 1,
we obtain a completely multiplicative, m-periodic function X
:
IN -*{ z E C, Izi = 1 or z = 0 },
defined by X(n) = X(n mod m) If gcd(n,m) = 1, and X(n) = 0 otherwise.
1.4. THE TURAN-KUBILIUS INEQUALITY
An additive function w: IN - C is called strongly additive if the values of w at prime-powers are restricted by the condition w(pk) = w(p), if k = 1,
2, ...
.
In 1934, Paul TURAN [1934] discovered the following inequality for the strongly additive function n H ca(n), the number of prime divisors of n: (4.1)
1 ((j(n) - loglog x )2 s c
nsx
x
loglog x
with some constant c. P. TURAN used this result to reprove HARDY
Tools from Number Theory
20
and RAMANL[IAN's theorem [1917] that ro(n) has normal order loglog n. Inequality (4.1) was generalized by J. KuBILIUS [1964] to additive functions, and later "dualized" by P. D. T. A. ELLIOTT [1979]. If w is strongly additive, then
Z w(n) = Z
Z w(p) = pox Z w(p)
[x/Pl,
nsx pin
nsx
and so w(n) is, on average, heuristically approximate to 2: p-<X p w(p), The so-called TUPAN -KuBILIUS inequality gives an estimate for the 1
difference of the values of the function minus the "expectation":
w(n) - Y_ p_'-w(p) psx
in mean square. its general form the TURAN-KUBILIl1S inequality has often been applied to the study of arithmetical functions. We use this inequality in Chapter VII in order to approximate functions in 11 by even functions In
and to outline criteria for additive
and multiplicative functions to
belong to 21q
For an arithmetical function w (and x > 0) we define the expressions A(x) =
(4.2)
Z P -k. w( pk),
P SX
E(x) =
(4.3)
P
and
P`sx
p
k' Iw(Pk)I2.
Theorem 4.1 (Tur'n-Kubilius inequality). There exist constants C1, C2 with the property that for every x2 2 and for any additive function w the inequalities (4.5)
xI
nsx
I w(n)
- A(x) 12 s CI D2(x)
and (4.6)
x
> I w(n) - E(x) 12 s C,-D2(X)
nsx
are true. In fact, it is possible to have C, = 30, C2 = 20.
14 The Turdn-Kubilius Inequality
21
Remark. If w is strongly additive, then the CAUCHY-SCHWARZ inequality gives
A(x) -Z p p-1'w(p)
I
I
s
s p.
P-1'Iw(P)I2)1
psx
P-2'Iw(P)I
p-k,I w(Pk )I 5 2 2:
k22
P:rx
(
2
psx
5 (1 psx
p-3)
P-1'Iw(p)I2).,
and
D2(x) s 2 E psx Therefore, from (4.5) we deduce (4.7)
x-1
I
nsx
w(n) -
P-1. w(p) I2
psx
s 2 x-I Ix
( 4 CI + 2
s
for
I w(n) - A(x) 12 + 2 I A(x)
-
psx
1 psx p-1 . Iw(P)I2
every strongly additive function w. Note that the constants are far
from being the best possible.
Proof of Theorem 4.1. Inequality (4.5) is a consequence of (4.6). By appropriate application of the CAUCHY-SCHWARZ Inequality we obtain E Iw(n) - E(x)I)2 x Iw(n) - E(x)I2 s nsX
nSX
and
s x-I
E Iw(n)-E( x)I2 + 2x-1 E Iw(n)-E(x)I
nsx
nsx
+
(
Ep
p`sX
p-k-I.I w(Pk )I)2 5 (C2 + 2 C2' +
-k-1 .I w(Pk)
1
I
D2(X).
)
We follow the proof given in ELLIOTT [19791, p. 148. There is another proof, due to ELLIOTT 119701, which uses the "large sieve". First, the assertion for complex-valued functions is reduced to the corresponding assertion for real-valued functions in an obvious manner, and then the
assertion for these functions is reduced to a problem concerning nonnegative functions.
(i) Assume initially that w is real-valued and non-negative. Then S=
nsN
(w(n)-E(N))2 = 2]
nsN
w2(n) - 2 E(N)
nsN
w(n) + N
E2(N)
Tools from Number Theory
22
= S1 - 2 S2 + N
E2(N)
say.
,
First, S1 =
GN psN
w2(n) _ 21nsN Xp`IIn w(Pk)
w2(P )
'
1
nsN
+
ZgQIIn W(qQ)
w(p k)
plq'esN p*q
p` Iin
w(q- )
.
a(N),
where, for distinct primes p and q, #(N)
I
Zn,N, p`IIn, gEIIn
counts the number of integers, which are exactly divisible by the prime powers pk, q1. Then
+
[ N/ p kqe+l
Pk+1'qt
u(N) = [N/pk,q.e ] - C N/
1
[ N/ pk+1,qt+1
(4.8)
_
N/pk.q'
-
1
1
P
-
)
q
+ 20,
where 101 s 1, and therefore
S1 s N
D2(N) + N ' E2(N) + 2
P*q
w(pk) ' w(qt)
Second,
EnsN'p`IIn w(pk) = Gp,sN w(pk) .
Lp-k.N] (
-
[p-k-1
N]
N - Ep.SN w(pk).
p-k, Z
Putting these estimates together, the term E2(N) cancels. Application of the CAUCHY-SCHWARZ inequality gives N-1 S s D2(N) +
w(p k) E p`gEsN, p*q
s It + 2'N-1 ( p'gQsN, P*q pk
.
w(qt) + 2-N-'- E(N) . Z
p`sN
q'e+ p`sN
pk E p kD2(N)
Some standard estimates complete the first part of the proof:
k p*q p
qe)S
w(Pk)
2
-r2-,
j.4. The Turn-Kubilius Inequality
Y-p.SN p
ks
23
41
ZpwSN pk S N'
nsN
n 1 S log N,
log-1(N) N a 2.
1 S 8 N2
The last estimate uses ZpSN log p s 2-log 2' N (for N 2 2 ). This implies PIN I + ZPwsN,kk2 I
S
+ ZpsN ( log W) -1
log p +
21ks log N/ log 2 psN,.` 1 s Ni + (2/log N) ' 2 log 2 ' N + ( log N/log 2 ) ' N21 S 8 N ' log-1(N). S
ps,/N
1
Thus the method gives the constant 1 + 2 ' 2 + 2'i S 10. Due to part (ii) this implies C2 = 20.
If w is
real-valued, define additive functions f+, f , where f+(Pk) = max (0, w(Pk) ), f (Pk) = - min (0, w(Pk) ). Define E+(N), ..., (ii)
D (N) in the same manner as E(N) and D(N), but now using the functions f+ and C. Then I
w(n) - E(N) 12 S 2 ( If +(N) -
E+(N) 12
+ If (N) - E (N)12 ),
and utilizing the relation D+(N)2 + D -(N)2 = D2(N), we obtain the result for real-valued functions, unfortunately with an additional factor of 2.
(Iii) The case of complex-valued functions is reduced to the real-valued case in the usual way by decomposing f into its real and imaginary part.
Next, we are going to "dualize" the TUBA N-KUBILIUS Inequalities. Consider the complex vector-space CM of all vectors 2 = (z1, z2, ..., zM) with Euclidean norm (4.9)
11211=(
and the usual inner product
.
For linear maps L: CM -4
CN
the "operator-norm" (4.10)
11
L II
=
11;2 11
=1
II
L(a)
II
is used. If, with respect to the canonical basis, the matrix C = (cm,n ),
Tools from Number Theory
24
1 s m s M, 1 s n s N, is associated with L, then the adjoint operator L* ( defined by < Lx, ti > _ < x, L*14 > ) is connected to the matrix Ct. Because of
II L II
=
II L
II we obtain the following result.
Theorem 4.2 (ELLIOTT's Dualization Principle). Let cm,n ' m =
1,
""
M,
n = 1, ..., N, be complex numbers, and let c > 0 be given. Then the inequality 2:.:,N I Zm:M Cm,nZm 12 5 C
(4.11)
ImsM IZm I2
is valid for all 2 e CM if and only if the "dual Inequality" (4.12)
Y-msM 1
2:nsN Cm,n Wn 12 5 C
2:nsN 1 Wn 12
is true for every to e CN.
Applying this principle to Theorem 4.1 yields the following theorem.
Theorem 4.3. For all x2:2 and for all complex sequences (w n ) the following Inequalities are valid: k'Ip
(4.5')
p°sx
(4.6'' )
GP
P
kP
x
pYsx
X nsx,p"IIn
1
- x nsx Z
wn
wn
W'(1-p1)' nsx nsx,p"IIn n x
12
C
l
2: x nsx
wn 2 5 C 2
1wn12,
1' nsxZ x
I W 12.
Further examples of applications of the dualization principle are given in the exercises, p.41. Finally, there is the following generalization of Theorem 4.1 to higher powers: Theorem 4.4 (TURAN-KUBILIUS-ELLIOTT Inequality). Given q
z
0,
there is a constant c > 0, so that the Inequalities x1
'
nsx
I
w(n) - A(x) I`' S c x1
I
nsx
w(n)
'
(Dh(x) +px I
P-k.
I
W(Pk)I9 ), If q > 2,
A(x) I4 s c' D9(x), if O s g s 2,
are valid for every additive function w and every x >-2.
The special case where q = 2 is Theorem 4.1 (only the numerical value of the constant c is not specified). We do not use this generalization,
and so we do not prove it, but, rather refer the reader to P. D. T. A. ELLIOTT 11980c].
I .S. Generating Functions, Dirichlet Series
25
I.S. GENERATING FUNCTIONS, DIRICHLET SERIES
The study of meromorphic functions near their singularities leads to arithmetical insight. In order to obtain meromorphic functions associated with arithmetical functions, different kinds of generating functions are used which are often treated purely formal; among the best known are LAMBERT series, generating power series and DIRICHLET series. (a) LAMBERT series: associate with a given arithmetical function
f; N -- C the infinite series L(f,z) = Znal
f(n)-z° (1-z°)-1.
Then, assuming absolute convergence in the [open] unit disc
B(0,1) = (z E C; Izi < 1),
the series L(f,z) can be transformed into L(f,z) _ Zn21 f(n). Zk20 zn(l+k)
rz1
z"
(
1
=
rx1 zr- Zdir f(d)
* f )(r).
Examples. In Izi < 1 the LAMBERT series L(1, z) _ n21 t(n) z° , since 1 * I = t , and L(X,z) _ n_t z°, where A is the completely multiplicative function taking the value -1 at every prime p. It is easily checked 2
that 1 * X = 1sq' the characteristic function of the set sq of squares.
If some suitable condition restricts the growth of the arithmetical function f, then the LAMBERT series L(f,z) is holomorphic in B(0,1). In = 1. This is true
general, there will be a singularity of L(f,z) at z
when the convolution 1 * f is non-negative and infinitely many of the values (I * f)(r) are non-zero, for example. Conclusions about the behaviour of the coefficients are often possible with the aid of Tauberlan Theorems; some of these theorems, important in number theory, are summarized in the Appendix (A.4).
Tools from Number Theory
26
(b) Generating power series.
We associate with the function f :IN0 - C the power series
9'(f,z) = nz0 f(n)
(5.1)
Zn.
If the function f is not too large, the power series (5.1) will converge in the complex unit circle B(0,1) = { z E C, Izi < I }. In order to obtain arithmetical conclusions, the most interesting singularity is generally the point z = (if this point is a singularity. This is certainly true, If f is non-negative and if infinitely many values of f are non-zero), However, in the case of the partition function n H p(n), for example, 1
with generating power series (S.2)
n=0
p(n) . zn = k=1 TI ( 1 - zk)-1
there are many other singularitites which have to be investigated if better estimates of the remainder term are desired. The method to be used is the analytic HARDY- LITTLEWO OD- RA MANUJAN circle method;
the coefficients p(n) of the power series (5.2) are expressed through a contour-integral, the main contribution to this integral being from small arcs of the integration path near the singularities of the function on the right-hand-side of (5.2). A useful device is outlined in HALL-TENENBAUM 11988]. If f Z 0, and 9D(f,z) converges in some interval I of the real axis including the point 1, then, obviously, for any N > 0, (5.3')
nsN nxN
f(n) s
f(n)s
f(N) s
inf x N . P(f,x),
O<xs1
inf x N x21,xcI
inf
x>O,xcl
x-N
9'(f,x),
P(f,x).
For example, using f(n) = (n!)-1, the last equation gives (N !)-1 s (e/N )N,
a rather good lower estimate of (N factorial). Another application of this principle is also taken from HALL-TENENBAUM 11988], section O.S. If E is a set of primes with least element
po(E), E(x) = Xpsx,pcE p-1, and Q(n,E) the total number of prime-
I .S. Generating Functions, Dirichlet Series
27
of n [counted with multiplicity] which lie inside E, then, for
divisors 0c< y < po(E ), the following Inequality is a consequence of II. Theorem 3.2: yf](n,E) du.
The integral defines a holomorphic function in Re s > 0, and so formula (5.9) provides an analytic continuation of C(s) into the half-plane Re s > 0, showing that c(s) has a simple pole at s = 1 with residue 1. Further integrations by parts of the integral occurring in (5.9) give the
analytic continuation of C(s) into the whole complex plane; which can be achieved in one stroke by the functional equation
(5.10)
c(s) = 2s
r 1,
where the VON MANGOLDT function A is given by (5.13)
log p
A(n) = l 0
5.000
,
if n is a power pk of the prime p, otherwise.
10.000
15.000
F i g u r e I.8. Primes in intervals of length 100
20 000
Tools from Number Theory
30
The number rt(x) of primes in the interval [1,x] behaves rather erratic locally. This is illustrated in Figure 1.8 on the foregoing page, giving the number of primes in intervals of length one hundred, from k s 199. The first interval contains twenty-five primes, the next one twenty-one, etc. , but there is also an interval containing only five primes.
The problem of obtaining an asymptotic formula for the number of primes up to x , rc(x) = > psx
(5.14)
1
,
is equivalent (via partial summation) to a suitable approximation to
,9(x) = I log P psx
(5.15)
or to (5.16)
4(x) = I A(n), nsx
via the easily verified relation [use the fact that higher powers are rare]
O(x) = E Mn) + of x' (log x)2 nsx
.
The function 4(x) = 1nsx A(n) has an integral representation (by a complex inversion formula), (5.17)
4(x) = (27d)-1
.
('c+i. C-1-
(-
s-1 x9 ds,
where c > 1. The "Method of Complex Integration" allows approximation
of the integral in (5.17) by shifting the path of integration to the left. The pole at s = 1 gives the main term x. Further contributions to the asymptotic
(- '(s)/i (s))
formula s-I xs,
follow
from
poles
the integrand which are caused by zeros of the RIEMANN
zeta-function in 0 < Re s
1.
Moreover the series Zn 1 is conditionally convergent in Re s > 0 if y is not the principal character. If X is the principal character Xo, then
j.7. Characters, L-Functions, Primes in Arithmetic Progressions
L(s, Xo) = 11
p.i'm
1 - p-S )-,
= pIm n (1 -
p-S l
l
37
c(s)
DIRICHLET characters X satisfy (like characters in locally compact
topological abelian groups In general, where summation is replaced by integration with respect to the HAAR measure on ;°) the orthogonality relations:
If a runs through a full set of representatives mod m ( for example, a = 1, 2, ..., m ), then X(a) =
a mod m
(7.6)
( cp(m),
If X is the principal character, otherwise.
0
If y runs through all the p(m) DIRICHLET characters mod m, then X(a) - J rp(m), if a = I mod m,
(7.7)
otherwise.
0
Corollary (Orthogonality Relations for DIRICHLET Characters). (7.8)
a mod m
X1 (a) '
X2(a) _
rp(m), if Xl = XZ, 0
otherwise,
and
9(m), if at = a2 mod m and I X(a1) x
.
gcd(ala2, m
X(a2) _ 0
) = 1,
otherwise.
These relations allow specific residue-classes mod m to be singled out: If f is an arithmetical function and gcd(a,m) = 1, and if n', are positive integers, then (7.10)
tsJ, nL=a mod m f(n) = 9(m) 1
x
X(a)
LSJ
f(n
I.
= 1, ..., J,
I.
Since
- L'(s,X)/L(s,X) = In= X(n)
A(n)
.
n-S
one finds results on primes in arithmetic progressions in the same way as is possible for ordinary primes (for example, using the method of "complex integration"). DIRICHLET L-functions have properties similar to those of the RIEMANN zeta-function, and so, using the functions
Tools from Number Theory
38
TE(x;a,q) = Ipsx, p=a mod q 8(x;a,q) =
(7.11)
1,
psx, p@a mod q log P,
(x;a,q) = nsx,
mod q A(n)
one obtains the following theorem. Theorem 7.1. If gcd(a,q) = 1, then, with some positive constant y, depending on a and q, the following asymptotic formulae hold : , 1
rp(q)
1q
(7.12)
cp (
)
1
x + C (x'exp( - y (log x) 2
) ),
,
x + O ( x'exp( - i (log x)2) ), Ii x + 0(x'exp( - y (log x)2 ) ),
W(q)
with the Integral-logarithm Ii x =Ii e + fe
(log u)-1 du.
It is sometimes important to have uniform estimates on n(x;a,q) in
I s a s q, with q restricted to some range, depending on x. An Important result of this kind is provided by the following theorem. Theorem 7.2 (Prime Number Theorem of PAGE-S!EGEL-WALFISZ). If I s a s q, if gcd(a,q) = 1, and if I s q s ( log x )A with some fixed constant A, then, as x tends to infinity, the asymptotic formula (7.13)
Tt(x;a,q
) = w(q) Ii x + 0A (
Y (log x)2} )
holds uniformly in a and q. As indicated, the constant Implicit In the 0-symbol may depend on A. For a proof see, for example, PRACHAR [1957] or ESTERMANN 119S21.
For some applications the range of admissible values for q in (7.13) is not sufficient, for example when consideration of larger values of q is unavoidable; this occurs in problems from the additive theory of numbers. The sieve method (V. BRUN, A. SELBERG) or the "large sieve" easily gives the upper estimate (7.14)
n(x;a,q) < Y '
x/(
p(q)log(x/q)
I.S. Exercises
39
With some constant y (which, in fact, may be taken to be 2, as long
as q
ao.
Then the mean-value M(f) exists and equals
M(f) = Z'=I n-1'f'(n)
(2.4)
Examples. (5) The convolution formulae µ2 =
(
(1101). sq) * 1, id/cp
where sq is the characteristic function of the set of n-Z squares of integers, lead to M(µ 2) _ Z n=1 µ(n) = 6 n-2, and to x-'- Xnsx (n/p(n)) -3 IT (1 + (6) The function a r(n), where r(n) is the number of representations of n as a sum of two squares, is multiplicative and representable as = I
* (µz/gyp ),
{(p-1)p)-1).
Eden X(d), where X is the non-principal character modulo 4; therefore r = I * X, with convergent sum X d-1 X(d). So AXER's result gives the mean-value result I
M(ar)=1-1
t
1
1
3
5
..=a>t.
Remark 2. WINTNER's Theorem ( Corollary 2.2) follows from AxER's Theorem: The absolute convergence of X-=1 implies the following by partial summation (see I.1):
Z n-.If'(n)I nsx
.
n = x- nsx E
('x z n-1.lf'(n)I du s C'x. i
nsu
Mean-Value-Theorems and Multiplicative Functions, I
54
In fact, the same idea and sensitive handling of the sum and integral yields the stronger result Z W(n)I = o(x).
nsx
Remark 3. Condition (2.9) alone is not sufficient for the existence of M(f). See, for example, A. WINTNER [1943].
Remark 4. If M(f) exists and If 2:n n-1 f'(n) is convergent, then M(f) = Z n
Proof. The existence of M(f) implies M(f,x) = Z f(n) = nsx Partial summation gives
£(f,s) =
M(f)
n
o(x).
(s-1)-1, as s -4 1+.
Therefore, M(f) = lims-->
1(s) = limes
1+
1+
But the convergence of Implies, by the continuity theorem for DIRICHLET series (or, what amounts to the same, by partial summation), that lim9- 1+ E and Remark 4 is proved.
Proof of AXER's Theorem. Abbreviating (i - [3] by (a), a routine calculation (change of the order of summation) gives nsN
f(n)=E nsN Edin f'(d) N ' Ed-M n n+1)
s N max M<nsN
`
M
N
For integers n ¢ T the inequality
1s n+f
I
n
IM(f', n)I.
Mean-Value Theorems and Multiplicative Functions, I
56
implies that there are at most M elements in J'; thus I{n-}-{n1I52
n 7
IR(N)I 5
s
IZdsM
f'(d)
a} I
6 NM max M<nsN
N
+I 2:M 0, then fix M = IM(f', n)I < E2 N. Then possible by (ii)) that max M<nsN
Is
11
11.3. ESTIMATES FOR SUMS OVER MULTIPLICATIVE FUNCTIONS (RANKIN'S TRICK)
This section deals with sums over non-negative multiplicative functions; the results given connect [estimates for] the size of values of f at prime-
powers in mean with the estimates of the sum Zn.,N f(n) from above (this Is rather easy) and from below (this is more difficult). The Ideas Involved are not too difficult: to obtain an upper estimate for EnsN f(n), where f z 0, an additional weight factor g(n) z I is introduced (for example, g(n) = log n If n z 3, or g(n) = (N/n)o if n s N), which makes the treatment of the sum easier (for example, log n can be split additively, or other factors g(n) make it possible to remove some troublesome conditions of summation without increasing the sum under consideration too much); the factor g(n) has to be chosen in such a way that the new sum EnSN f(n)-g(n) can be dealt with in a simpler way. Surprisingly that this simple method ("RANKIN's trick") is often very effective. Theorem 3.1. Suppose that for a non-negative, multiplicative arithmetical function f: N -> [0,co[, for every y z 1, the upper estimate
0 3. Estimates for Sums over Multiplicative Functions
P2]
S7
y (log Y),
f(pk) log pk s C
is true with some a 2 0 and some positive constant c1. Then there are constants c2, c3, which depend only on c1 so that for all x Z 2
Zx f(n) s c2 x (log x)'-' exp f p x P-1. f(p)
(3.2)
(3.3) Zx f(n) s c3 x log°`x exp { p21x p
+pZ k 2
P-k f(pk) },
E k2 P-k. f(pk)}. psx
Remark. If, for every prime-power pk, 0 s f(pk)s Y1 Y2k, where Y2 < 2, then (3.1) holds with a = 0.
Proof. The function n H log n is additive, and so log n = p"Iln log pk Inserting this into Y-nsx f(n) log n, inverting the order of summation, and using the multiplicativity of f, we obtain
I f(n) log n = p"sx E log pk E f(m) f(pk). nsx msx/P", P-l' M Inverting the order of summation again, and neglecting the condition this leads to the estimate Z f(n)-log n s 21 f(m) msx
nsx
p
"
7-
sx/m
f(pk) log pk s ci x (log x )°C
Z M-1-f(m). msx
Assumption (3.1) was used in the last line. The multiplicativity of f and the inequality I+ y s exp(y) imply
I m-1 f(m) s p11
msx
s exp
l (
+
I
P-1, f(p) + p
...
}
Z P-k' f(Pk)) = E(x), psx ki2 for the moment. Then, for x z 2, E f(n)-log(n) s ci x log°` x E(x), and
I
2snsx
Z psx
nsx
f(n) s s c1 x log°'-ix E(x)
n/ log 2) + { X-21
E
I<nsx
f(n)-log n/(2 log x)
log x/ log 2 + 2} S 4 c1 x logac-1 x E(x).
In our argument we used the fact that y H E(y) is monotonically increasing, and that the maximum of x- log x / log 2 in x 2 2 is attained at x = e2 and equals 1.06 ...
.
The estimate Zp5x p-1 s loglog(x) + c4 (see 1.7) implies the second assertion.
Mean-Value-Theorems and Multiplicative Functions, I
58
Another form of this theorem is given in G. HALL & G. TENENBAUM [19881, Theorem 0.1.
Theorem 3.2. Let f be a non-negative multiplicative function satisfying (3.4)
p s cIX ( X 2 1
P:CX E
),
and
log(pk) 5 c2.
Z P-k.f(pk) p k22 Then, for any x Z 1,
(3.5)
Z f(n)s ( c1 + c2 + 1) x
nsx
Z n-'-f(n). nsx
log-1 x
Remark. WIRSING'S condition f(pk) s YI'Y2k for k z 2, where 0 s Y2 y, and that for some A > 0 the series
0.3. Estimates for Sums over Multiplicative Functions
p Yk22
(3.6)
S9
pk. f(pk). pkA
is convergent. If
log( log z/logy / log y --> 0
(3.7)
as y -- oo, then
log z [ E P-'-f(p) -log( llog z_)]). log y p5y
(3.8) n>z,P(n)sy n-'-f(n) x,P(n)sx E µ2(n) n>x,P(n)sx
µ2(n)
weight-function neglecting the condition
with
again,
idea
log n /log x, this difference is (writing n = d i x/p and using h(p) s 0.1 µ2(n)
61
(
1.SS
log x)
1-1 (
1
2(d) d-I h(d) 1
+
p-1. h(p)
+ P-1 h(p)
if x 2 e3, and the assertion will follow, after replacing 1.55 by 2. Theorem 3.5 [BARBAN]. Let g be a non-negative multiplicative arithmetical function bounded at the primes, 0 s g(p) s C1.
Then there Is some positive constant C2, depending only on C 11 such that the Inequality (3.11)
n
Z
r
µ2(n).n-1.g(n)
2 C2
exp ( N P-1'g(P) P
holds as soon as N is sufficiently large.
Proof. In order to apply Lemma 3.4, choose an integer m so large that m = [10-C I I + 1, and put z = N1/r". Define completely multiplicative functions g* and H0 by (3.12)
g*(Pk) = {g(p)}k, k = 1, 2, ...,
H0(n) =
m-0(n)
.
n-1 .g*(n)
If H is any non-negative, completely multiplicative function, then
Mean-Value-Theorems and Multiplicative Functions, 1
62
H(n) }m = Zz ,..., n . H( n s rszm Z H(r) t m (r)
with the divisor function tm(r) counting the number of representations
of r as a product of m factors. The values of tm at primes p are tm(p) = m, and thus tm(n) = m0(n) if n is squarefree. Using the representation tm = 1 * ... * 1 = 1 * 'cm-11 the relation (3.13)
tm(Pk)
k+m -1 =l m- 1 ),k=1,2,...,m=1,2,...,
is easily proved by induction (Exercise 16).
Write r in the form r = r'-d, where d is squarefree, r' is 2-full (this means that pir' implies p2Ir' ), and gcd(r',d) = 1. Then, neglecting the condition gcd(r',d) = 1, we obtain
Ensz H(n) }m s Xr' szm, r' 2-full H(r) tm(r') psz^'
1
2: dSzm
H(P3)'tm(p3) + ...
+
Ed:,.,,
x
}
i
With the choice H = HO given above (see (3.12) ), and paying attention to g* (p) s C1, the product PO = 1f 1 + HO(p2).tm(P2) + HO(p3).tm(p3) +
...
}
is convergent (for this, an estimate such as tm(pk) «m'e pk is useful). The sum G(N,m) = { nsN-
j2(n).n-1,g*(n),m-n(n) }rn
={E
O (n)
nsNvm
satisfies
E µ2(d) H0(n) }m s P0 d sN G(N,m) 5{nsN"m E
.
d-1, g(d)
on the one hand; on the other hand we obtain, from Lemma 3.4, G(N,m) 2
( 0.8 )m , n pSNvml
These two estimates Imply the relation
1
+
g(p) )m.
}=n
19.3. Estimates for Sums over Multiplicative Functions
dsN
µ2(d)
d
(0.8
g(d) z P o
1
)m P
n (I +
63
\n,
p
m
l
for every N 2 e3m. For 0 s x < 1 the inequality z (1-x2
(1+x) = (1-x
is valid. Thus we obtain (with x = g(p)/mp)
TI (1 - (
P1
)2 ) > IT
(
P
P
1 - 0.01. p-2 1J > 0.9
(by an easy numerical estimate), and, finally, d
µ2(d)
N
d
z po
-I.p1m.
( 0.8 )n'
.
exp(-C1
x exp ( psN E
N
psN p
1
m
The well-known asymptotic formula for Zpsx P-1 (1.6, (6.7)) yields N ,gym 0 the asymptotic relation (4.1)
t + 0(1))
p
psx
x, x --> co
holds, and that for every prime p and k = 2, at prime powers are 'small', f(pk)
(4.2)
3, ...,
the values of f
sY1Y2 k, where 0< '2 oo, the asymptotic formula
(4.3) nsx Z f(n) _ (1 + o(1))
e-et
x
log x
I
-,
I' (t)
T1 psx l
+ f(p) + f(p2) p
p2
+
...
holds. i° denotes EULER's constant, r(.) the gamma-function. Remark 1. (4.2) may be replaced by weaker assumptions, e.g.
of (log x)-1 ),
E P Zk=2, P"2x and
f(p) =
O(pl-g
) for some 8 > 0.
Using the Relationship Theorem from Chapter III (Theorem 2.1), these assumptions can be weakened further.
Mean-Value-Theorems and Multiplicative Functions, I
66
Remark 2. (i) Starting with (4.1), partial summation gives (4.4)
log x,
P-1'f(p)'log p = ( t + o(1))
psx
and, furthermore, the convergence of the series (4.S) (Flog P)-1 f(p) = f I(t + o(1)) 2 + log u du .
P
u
2
p = O(p), and this estimate, together with (4.S),
(ii) (4.1) gives
implies the convergence of
Z P-2' f 2(p) K in S"2, and use f(pk) 5 YIY2k. Then S"2 5 upsxv,,. , f(p) log p
0,
g(p) log(p)
g2(p)
(4.11) P
and, for all primes p and k = 2, 3, g(pk)
(4.12)
...
,
5 Yl'(Y2/P)k, where Y2 < 2.
Then
(4.13)
n 21
g(n) _ (1 + 0(1))
I'(t+1) r -L°t
_ (1 + 0(1))
1'(t+1
IT (1 + g(p) + g(p2) + ... psx '
P
exp (
psx g(P)),
)
11.4. Wirsing's Mean-Value Theorem
69
with a convergent product
P = II exp -g(p))'
(4.14)
(1
+
g(p)
+
g(p2)
+
... )
.
P
Remark 3. Condition (4.12) may be replaced by the weaker assumption
E E g(pk) < oo. p ka2
(4.12')
proof of Theorem 4.3. Consider the generating DIRICHLET series
Vg,o) _ Yn=l g(n)
n-O.
The following conditions hold: (i) (ii)
(iii)
The product P is convergent. £(g,o) is convergent in o > 0. o-t B(g,o) ti P exp ( H(o)
),
o -* 0+,
where
H(o) = Y_ g(P)'P ' - t
log(o-1).
(iv)
The function x H L(x) : = exp ( H(1/x)) is slowly oscillating.
(v)
Z g(p) P
(vi)
For any real r, 0 < r
0.01. It was produced using the com-
puter algebra system RIEMANN II (Begemann & Niemeyer, Detmold).
2 F i g u r e
11.4.
Therefore,
I
hp(a)
I
S kk2 g(pk) + 2 (k 1 g(pk) )2.
By the WEIERSTRASS criterion, 7,2(6) is uniformly convergent in 6 Z 0, and so 7,2(0) = 7, 2(0) + o(1), a - 0+.
7,1(6) is to be treated by partial summation. Split this sum into 7,1(x) _ Ipsexp(1/o)
g(P)'P-o
7,11(6)
+ 'p>exp(1/o)
+
7,12(6).
The second sum is easier to handle: 7,12(6) =
Je /6 { 1/aE
g
By assumption (4.7) the expression in braces {
...
} is
t
(log u - log exp (6-1)) + o(log u), and thus, using the substitution w = 6 log u, a straightforward calculation results in 7, 12 ( 6 )
t + 00) )
fa` 1
w-1
dw.
Mean-Value Theorems and Multiplicative Functions, I
72
For 211(0), we apply partial summation to the difference: P-°
(0) 11
psexp(1/°) g(P) log p
psexp(1/0 g(p) _
1/0
_
f e PsU I
pse 1/0
log p d P'u( lo°g u)du.
2
p, and sub-
Using the asymptotic formula (4.7) again for Epsx we obtain stituting w = 211(6) - Zpsexp(1/°) g(p) = O(1) -
[
f 1 w-1,(1-e-W) dw. O
Summarizing, using the integral representation 1
dw - f
L° = f
0
OD
dw,
1
for EULER'S constant, we obtain the formula g(P)'P_°
= 2:psexp(1/0) g(p)
Zpsexp(1/o)
+ 0(1)'
and (v) is proved. The function x H L(x) = exp ( H(x-1)) is slowly oscillating: without loss
of generality, c s 1; then, using the definition of H(.), (v) and (vi),
log L(x) = H((cx)-1) - H(x-1)
log
exp(x) 0, t > 0, and
2 If(Pk) I < ao; p k22 P-k ' then, in addition, the condition
(5.3)
If 0 < t s
1,
E
(5.4)
p kk2, p`sx
If(Pk)I = C7 ( x/log x).
is assumed to hold. Then (5.5)
21
nsx
f(n) _ (1 + o(1))'
e
log x
-°t
r(-0
. IT
+
f(p
psx l
f(p2)
+
+
p
P
...
.
Theorem 5.2. Assume that f: N -4 C is multiplicative and satisfies assumptions (5.1) to (S.4) of Theorem 5.1, and, moreover, that Z
(5.6)
p-I
.
(
If(P)I - Re (f(p))
)
< co.
P
Then assertion (5.5) of Theorem 5.1 is true.
Some preparations are helpful for the formulation of the next result. Let
g={ z= p'e1" E C; O s 9< 2n, O s p s r(p) } be some region in the complex plane C containing 0. Define its "average radius" by
r(g) _
(2n)-I
.
f2n r(y) dp. 0
Theorem 5.3. Assume that f: N -) [0,co[ Is multiplicative and satisfies the assumptions of Theorem 5.1. Let f : N -4 C be multiplicative, If s f. Suppose there Is a convex region g c C with average radius r(g) < 1, containing 0, such that contains all values f*(p). Then I
fz
nsx
(n) =
x
log x
e->°t
r(t)
rr / psx l
1
+
(p) P
+
f (p?) p2
+
...
+o
nsx
We do not prove these theorems here, but refer to WIRSING'9 paper [1967]; a proof by A. HILDEBRAND for a special version of WIRSING'S
Theorem for real-valued functions is given in Chapter IX, and, in the
Mean-Value Theorems and Multiplicative Functions, I
78
same chapter, we give a result due to G. HALASZ1) [1968] for complexvalued multiplicative functions of modulus IfI s 1; the proof given there will be "elementary" and follows H. DABOUSSI and K.-H. INDLEKOFER [19901.
Theorem S.4 (G. HALAsZ). Let f be a multiplicative arithmetical func-
tion of modulus IfI s 1. Then there exist a real constant a, a com-
plex constant C and a slowly oscillating, continuous function L: [1,co [ -) C, ILI = 1, for which the asymptotic relation
Z f(n) = C
nsx
L( log x )
xl+Ioe
.
+ o(x)
Is true.
The function L and the constants a, C may be given explicitly. 2) proof of parts of Theorem S.4 is postponed until Chapter IX.
The
11.6. THE THEOREM OF DABOUSSI AND DELANGE ON THE FOURIER-COEFFICIENTS OF MULTIPLICATIVE FUNCTIONS
In 1974 H. DABOUSSI and H. DELANGE announced the result that, for irrational values of a, FouRIER-coefficients f^(a) of multiplicative arithmetical functions f of absolute value IfI s are zero. DABOUSSI and DELANGE [1982] proved the following stronger result. 1
Theorem 6.1. Let f be a multiplicative arithmetical function for which the semi-norm (6.1)
1)
2)
IIfII 2 2
: = lim sup x --> co
x-1
' nsx E If(n)I2
In the authors' opinion, GABOR HALASZ's method, a skilful variant of the method of complex Integration, seems to be definitely simpler than WIRSING's method of dealing with convolution Integrals. See also the paper by K.-H. INDLEKOFER [1981a].
II.S The Theorem of G. Hala sz
79
Is finite. Then, for every irrational a, the mean-value (FouPJERcoefficien t)
f (a) = M(
(6.2)
lim
x - eo
°C
x-1
E f(n) e2"Ian nsx
is zero. We do not give DABOUSSI and DELANGE's proof, but sketch a proof of a result which is a little weaker - the relationship theorem of Chapter III allows the deduction of Theorem 6.1. The result is as follows.
Theorem 6.2. Denote by TA the set of multiplicative functions with the properties If(p)I s A for all primes p,
(6.3' )
(6.3")
n
N If(n)12 s
for all Integers N Z 1.
Abbreviate by Sf(c() the exponential-sum
S (a) _ z. f
nsN
Then S() = o(N), as N - co, If f e S'A and a Is irrational. Remark. Based on the "Large Sieve", H. L. MONTGOMERY and R. C. VAUGHAN [1977] prove a stronger result:
If f Is in SrA, then, for q s N and gcd(a,q) = 1, (6.4)
Sf(a/q) y if p 5 y
sY(p) -
n-1. s (n) = IT ( PsY
Y
1
1,
if p 5 y
0,
if
+ p-t + p-2 +
>
P
... )
y
< oo, and the
mean-value
M(Z) = M(l * s -t( ) = M(1 *(its ) ) _ (7- n Y Y Y >
(n) )-t
1.s Y
exists. This is proved using WINTNER'S Corollary 2.3, for example.
Define aY = (s
Y
Y
Y
(n/d). Then the mean-value
M(( ) = (EbcR b-t,s(b) )-t
(8.4)
co, this follows from Theoexists for every y 2 2. Since Z rem 2.4 for example, and, by multiplicativity, the result n-I.sY(n) (
)-t
. C n-t
,
(a'sY)(n)
is easily transformed into (8.4). Next we show that
Mean-Value Theorems and Multiplicative Functions, l '
84
M(a,x)
(8.5)
Gnsx a(n) s M(oeY,x), for x z
:=
1.
Taking this for granted, the theorem is proved as follows: a*a = plies lbc.8, bsx M(a, x/b _ Ensx 1 = [x], and so M(a,x) +
bc.X3, 1 oo
be.
b-1)-1
be.8
It remains to prove lnsx a(n) 5 Z.!x aY(n) (this
is (8.5)). Since, by
complete multiplicativity, the relation sY (a (3) = sy, a * sY p holds, we s obtain s Y
X a(n) = nsx
a(n)
(
s a*s a * Z Y
nsx
Y
)(n)
Y
Eabc&x,acA,bc.s s (a)
y
csx/a
$
y (c) 21bsx/ac,bcR sy
5 lasx,ac.r1 sY(a) Ccsx/a Zy(c) = Here
1nsx(sY'a*tY)(n)
= M(ay,x).
1 1.9. Daboussi's Elementary Proof of the Prime Number Theorem
s
Zbsx/ac,bc2
8S
s 1,
used. This is true, since, given a number of the form ac, there is A e s4 [otherwise, if b'-ac = A' a s4, at most one be. for which A' b, which is impossible, since IN is a direct prob' $ b, then was
duct of .vl and B ].
11.9. DABOUSSI'S ELEMENTARY PROOF OF THE PRIME NUMBER THEOREM
The Ideas used In 11.8, worked out in greater detail, make It possible to give an elementary proof of the prime number theorem (DABOUSSI [1984]).
Theorem 9.1. Put M(x) = Y-nsx µ(n). Then lim
x -- m
x-1
M(x) = 0.
Proof. With some [large] parameter y we use the completely multiplicative functions 8y, sy defined In 11.8, where ty(p) = I for "large" primes p > y and sy(p) = 1 for "small" primes p s y. For brevity, write M (x) = nsx µ(n) s (n). Then µ = µE *µsy, and so y
y
y
M(x) = 7nsx
y
(n) My (x/n).
the finite sequence of squarefree integers composed only of prime-factors p s y; in x/d +1 < n s x/d the I function My (x) is constant. Therefore, with d q+1 = oo, x/dq+1 = 0, we n Denote by d1 = 1 < d2
(1-20)-1. We apply inequality (9.7) in the interval a s t s a and we apply in [xp s ] the estimate IM(t)I s s t s b. This leads to
fab t 2
IM(t)I dt s 2P
log(1-2P)-1
+ P log((b/a) (1-2P))
= P log(b/a) + 0 log(1-2P).
Using the definition of S and 2P log(,-2, 3) 5 - a P2 5 - : a2 we see that
2P log(1-2P) 5 - 6 (8-1) 5 - (1-5-1)
.
P
log(b/a),
and so we obtain (9.6) in the third case also. Proof of (9.4ii). The following auxiliary functions are needed: (9.8)
F(x) = fox U-1
(1-e-u) du, where x > 0,
Mean-Value Theorems and Multiplicative Functions, l
88
k(s) = fo e-ax , eF(x) dx, where s > 0.
(9.9)
It Is obvious that the function s '- k(s) Is positive, decreasing and continuously differentiable. Furthermore, fss
s k(s) -
(9.10)
1
k(u) du = 1 for any s > 0.
[The left-hand side of (9.10) equals - f 0co eF(x)-sx
dx f Oco -A-
(F'(x) - s) dx
e
F(x)-sx
dx = 1.
I
For fixed y 2 2 we consider the function h, defined in x > 1: 1O
y)-1
1
kl
h(x) _ (log
(9.11)
Then, for any x > 1, (9.10) leads to the equation log x
(9.12)
h(x) = 1
Jxyx +
u-1
h(u) du.
Partial Integration gives (9.13)
5"
x-1
(
fxyx
u-1
h(u) du ) dx = f z (2-u)
k(u) du
log y.
Lemma 9.2. Denote by C the limit given in the displayed formula Immediately following (9.3). Then
k(u) du = C -
f12 (2-u)
1.
Proof. Starting with the convolution relations log = A* 1 , ands
y
and using the abbreviation S y(x) = nE
Sy(x)
sy(n)
log = ( sy A) * s y ' n sx
sy (n), we obtain
log n = d sy(d) A(d) :cx
'
Sy l
d
)
log x = -x sy(n) log n +n-7 sy(n) log(x/n)
2
PSy.P sx
yp
nsx y
log X. n
II.9 Daboussi's Elementary Proof of the Prime Number Theorem
89
Therefore,
lo-Px h(x) dx = 27
(9.14) fy SY(x)
log p SY(
P)
hXX) dx + R1 + R2,
with the remainder terms Rl =
(9.14.1)
f-y E
/ xk) h(x) dx, x2
log p
P&Y
Syl p
k> 2, p`S x
R2 = fY
(9.14.2)
log(n) hXX) dx.
sY(n)
The error terms R1 and R2 are bounded: the estimate ' J1
u-2.S(u)dus nsx I n-1s(n)_ IT PSY Y
(1-p-1)-1=O(logy)
Y
Implies
Rl s h(y)
p-k log p , f °° u-2
7-
P, k>2
1
SY(u) du = d(1),
and
R2 = C'0 ( h(y)
I n_1 n-1 sY(n)) = O(1).
Starting with the elementary relation nSY p-1 log p = log y + 0(1) [see I. (6.6)], partial summation yields the formulae (9.15.1)
(9.15.2)
p-1 log p
P E
Y/t 1. Y
The Integral on the right-hand-side of (9.14) equals Z log p f m... dx, and using the substitution x H p
fy//P
t-2 h(pt)
SY(t)
p-1 dt
for this integral. Interchanging summation and integration, we arrive at
fYm SY(x)
12JIyx
Y h(x) dx = fY t 2 S (t) ft
+
f' t -2.S y
(t) Y
Y/t N. Prove that there exists a complex constant c Icl = 1, such that f is in TI and has the property ,
log-1
lSf(a) l >> N
N [see Theorem 6.2].
IS) Prove:
a)
cp-1
'
b) nsx c)
id = (cP-1 . µ2) * 1. (p(n)-1
n )2 = C) (x ). nsx n cp-2(n) = C'1( log x ).
Mean-Value Theorems and Multiplicative Functions, I
96
16) Prove the formula tm(pk)
k+m_
m -1
1
k, m = 1,
2, ...
.
17) Prove by mathematical induction the inequality n p < 4N. psN
l is odd, use n p s ( 2kk+ 1 psN
Hint: if N = 18) In 1 s a < b, prove
)
rI
P :r k+1
p.
t2 Gnstµ(n)dtl 56.
Hint: use I. Cor. 2.5.
19 ) For positive real numbers ai, bi, where 1 s
i, j
s n, calculate the
determinant
Dn = det(a + b
1s1TI n
(a, - a l
) (bt - bj) ( IT (ai + b
Y
Hint: subtract the n-th row from another one, and extract suitable factors. Then subtract the n-th column from another one. Proceed by induction on n. 20) Prove formula (10.3), using STIRLING'S formula
log(n!) = n -log n - n + O(n).
97
I
0
Chapter III
Related Arithmetical Functions Abstract. The simple fact that multiplicative functions are determined by their values at the primes leads to the Idea that multiplicative functions which do not differ "too much" at the primes behave similarly. The aim of this chapter Is to render these vague Idea more precise and to provide a universally applicable result In order to reduce proofs to the simplest possible assumptions. The notion of "relatedness" Is a measure for "not differing too much at the primes". Our result states that two related functions f and g, which are not too large, are con-
nected by a convolution formula g = f * h, where the function h Is small In the sense that the series Z lh(n)l ' n-1 is convergent. This chapter is close to the paper HEPPNER & ScxwARZ 119831.
Related Arithmetical Functions
98
III.1. INTRODUCTION, MOTIVATION
Multiplicative functions are determined by their values at the primepowers pk for the relation f(
IT pk) =
f(k)
J_ Pk1In
PkOn
Higher prime-powers pk, where k 2 2, are rare: the number of these
up to xis 1 = C7(xg,),
pksx, kk2
and so one is inclined to conjecture that multiplicative arithmetical functions behaving similarly at primes have similar properties". This chapter aims to give an exact meaning to these vague formulations. The theorem we are going to prove will be Important for simplifying proofs by reduction of these to special cases which are easier to handle (for example, multiplicative functions may be replaced by com-
1)
For example, given two multiplicative arithmetical functions f and g, which behave similarly at the primes, one might ask for conditions that ensure one or more of the following assertions: If f has a mean-value M(f) = Urn x-1 E f(n), then g has a meanx->
nsx
value, too.
If f has Fourier-coefficients f ('y) = M(
then the Fourler-coeffl-
cients of g exist. If the RAMANLUAN-Fourler-coefficients ar(f)
do
_
exist, the same is also true for the function g. If the RAMANLUAN-expansion 2.1sr<m ar(f) cr(n) of f is Cpolntwise, absolutely, uniformly, ... ] convergent, the same is true for g. If the series E1sn 2, and k =
1,
-1 , 0 ,
in
,
if k z 2,
is related to g, and is in but not in An easy computation shows that f-1(*)(pk) = 1, if p $ 2 and k is arbitrary. For p = 2, however, we obtain f-1(*)(2k) = 2k; therefore
h(pk) = f-1(W)/p ) - f-1(*)(pk-1)
=
2k-1
and so I n-1
Ih(n)I
if p = 2,
is divergent.
However, in spite of this example, a condition weaker than f sometimes sufficient for applications.
E
§°*
is
Related Arithmetical Functions
102
Theorem 2.2. Assume that f and g are related, and that both are in Assume, furthermore, that the factors Epf(p,s) of the EuLER product of the DIRICHLET series D(f,s) are not zero In the closed half-plane Re s Z 1 for every prime p outside some finite exceptional set h°r,
Then there exists a multiplicative function h in .AL°Ai (see (1.s)) satisfying g = f * h, provided f(pk) = g(pk) for every exceptional prime p e 9? and every k = 1,2,
...
.
m
Theorems 2.1 and 2.2 are deduced from the following theorem. Theorem 2.3. (1)
The set § Is closed with respect to convolution: f*g Is in ;, if f and g both are both In
Figure
III.1
.
(2) The set §* is closed with respect to convolution.
(3) If f is In §°*, then the convolution Inverse f-'(*) Is in (4) The set AeAi of functions f with absolutely convergent series n 1'If(n)I and the set RE of functions In §, related to the unit element s, are Identical. An extension of Theorem 2.1 to functions that are related in some (apparently) more general sense is easily possible.
Given p, 0 < R s 1, the multiplicative functions f and g are called 0-related, if ZP P-13-1 f(p) - g(p) is convergent. In analogy with notation given earlier we use the abbreviations I
(2.1)
.A 'J1ip =
(2.2)
Re0 =
{
{
f f
E
Ai; X n a If(n)I < co
E ;p; f a-related to E
{f E At, Z p- 213 If(p)i2 < m and E
(2.3)
},
p kl. If(pk)I
P0, (3.3)
0
,
if
p
sP o.
The function fo is completely multiplicative. The function defined next, f1' , is 2-multiplicative and inverse to f with respect to convolution:
Related Arithmetical Functions
106
f1' (p k )
(3.4)
- f(p), if k= land p > Po, =
otherwise.
0
The "tail" of f is defined as follows: fl..(pk)
(3.5)
f(pk), if p > po,
_
j
0
,
if
p s P0'
,
If
p > P0,
if
p
Finally, the "head" of f is
0
f2 (pk) =
(3.6)
We define f1 = f'1 * f1"
.
f(p
PO .
Looking at the generating DIRICHLET series,
it is obvious that f = f0 * fl * f).. * f2, and
f0-1(*) = f1,
.
the second assertion can also be seen from the relation
h-1(*)
= µ' h, which is true for completely multiplicative functions (see Exercise 1,8). C) The Main Lemma
then the following assertions are true: Lemma 3.3. If f, g (a) f*gE If f and g are in §.w, the same is true for f * g. (a*) (b) f0 ;'", and f -1(*) E*. .
E
(c)
f1" are in §, f1 is In .pI1'I1 In fact f1', fl" are In I ,P4L'It, where f1 = f1' * fl"
f1',
(f)
where f1 = f1' * f1".
f1-1(*)
(d) (e)
0
E
f2 E .a4L°AZ. If, for every
prime p s P0, Epf(p,s) * 0 in the half-plane
Re s Z 1, then f -1(") Is In 46AI.
Proof. Recall that §, § , p f(p,s) and AeAl are defined in (1.2) to (1.5). (a) For the moment we write w = f g. Then w(p) = f(p) + g(p), and so Ep-2'
If(p)I2+Ig(p)I2} 0at x=2, and has a unique local maximum at 2'-In 2 = 0.346... ; therefore (see Figure 111.2) 0,1
0,2
0,3
0.4
0,5
Figure 111.2
0.6
0,7
I
Using this inequality, the multiplicativity of f1 and the values fI(pk) given in (c), we arrive at the lower estimate
111.3. Lemmata, Proof of Theorem 2.3
p>P
{I -Z
Pk'(If(Pk)I+If(pk-1
k22
exp { - 2
>
109
S p> PO
P
}.
for all s in Re s z 1. Making use of the fact that f is in ; (and using the assumption If(p)I/p < 1/6 as long as p > Po), we obtain s
S
P>P0
P
P-2 If(p)I2 +
E
P> P0
Z Pk
L 6
p>P0 k22
if(pk)I 5 Y1
with some constant Y1, and so
n-' - ft(n) Iz8 =exp(-2Yt) inRes> 1.
IL.
An
application of Lemma 3.2 now gives fl-100C
the
desired
conclusion
,46m.
Proof of Theorem 2.3. Assertions (1) and (2) are already proved (see Lemma 3.3, (a) and (a*)). For (4), the assumption h c A1?A implies E p-1 Ih(p)I S Zn 1 n-t.Ih(n)I
x
dz1
Enlarging the first remainder term by multiplying every summand with )''-a (which is greater than 1), we obtain the assertion the factor (d/,, of Theorem 4.1 in the case where R(x) = O(xa).
Similar calculations allow to derive the result
in
the second case
R(x) = o(x13).
Example. If f c ;* has a mean-value, then the function f = µ2 f is in and related to f. Thus it has the mean-value (see (4.4)) M(i12f) =
1+p-1.f(p)) .(Xk2o
P-k.f(pk)1-1
Corollary 4.3. Let r > 1 be an integer, and f a multiplicative function, uniformly bounded at the prime powers, and let g be a multiplicative function such that the series P-k, I g(pk) I r E ka2 Is absolutely convergent. Finally, assume that f and g are related, and that
(4.5)
(4.6)
E
p
21
P
p_'-1 f(p) - g(p) Ir
Is finite. If, for every prime p, 9fr(p,s) $ 0 In the half-plane Re s 2 1, then the existence of the mean-value M(fr) Implies the existence of M(gr).
Proof. The boundedness of f at prime-powers and the condition on Pfr(p,s) show that the power fr is in ;*. Next, having shown (from (4.6) and (4.S)) that gr is in b°., and that fr and gr are related, then Theorem 4.1 gives the assertion. Only the proofs of (4.7)
1 P-2.Ig(P)I2r P
< ao
III.4 Applications
113
and of
X p-1 (4.8)
Igr(p) - fr(P)I < ao
.
P
are not quite so obvious. Firstly, g(p) = C)(p1'r), by (4.6), and, since If(p)I s K for all primes, using Ig(p)I S 2. Ig(p)-f(p)I, if Ig(p)I 2 2 - K, (4.6) implies that the sum p-1
zp,Ig(p)1-2K
.
Ig(p)Ir
o 1
0
folexp(-2nia q ) X-1- Z e27u1«A(n)d« nsx
exp(-2nia q ) lim
X -3
x-1,
E eZ nsx
,
b0 Mn) d«,
and thus the densities 8 q do exist.
11
The calculation of these densities is laborious. We perform this calculation in the special case of a strongly additive function A(.); in this case A(pk) = A(p) does not depend on k. Firstly, by Theorem 4.1, the mean-
value of the function f: n H exp(2nia A(n) ) is 1 + P-1.( e27t1«A(p) - 1
M(f«) _ The density 8
P
q
is then
)
) _ Z n=1 n-1' (v*f«(n)).
111.5. On a Theorem of L. Lucht
S
q
=
fo exp(-2nia q)
=
f
1
O
fo
1
115
M(f.') da
exp(-2nia q
)
G
(n
exp(-2nia q
I
(
)
II (1-
e2n1«A(p))lda l
pin
ln
J/
dn
Interchanging the order of summation and Integration (this is possible by the dominated convergence theorem) we obtain Sq =
din,
(d) = q
d-1,µ2(d)
m-1,µ(m)
m,(m,d)=1 plm=: A(p)*O
d, A(d)=q, pld=> A(p)*O
= IT (1-pA(p)*O
I )
(d)
I
d-1 µ2(d)
d, A(d)=q,
IT( 1-p
1)-1
pid
pld=> A(p)*O
III.5. ON A THEOREM OF L. LUCHT
Given q 2 1, denote by £q the C-vector-space of arithmetical functions f: IN - C with finite semi-norm (S.1)
IlfIlq = { Ilmsup x-1
E If(n) Iq }
1/q.
Note that fq is not closed with respect to convolution. For example, the constant function 1 n H 1 is in A1q = ?q n AI, where Al is the set of multiplicative functions. But, for any q > 1, the divisor function t = 1 s I is not in AIq. By contrast, (;,*) is a semi-group with identity element e, and (.*,*) is a group. :
L. LUCHT [1978] proved the following theorem, which may be considered
Related Arithmetical Functions
116
an important step towards Theorem 2.1. Theorem 5.1. Assume q > 1, and let the multiplicative functions f and g
In 9' be related. Assume, further, that, for every prime p, the factors cpf(p,s) of the generating DIRICHLET series for f are non-zero in the half-plane Re s z 1. Put h = f-1(*) g; then the series n-1 n-1
h(n)
Is absolutely convergent. In Re s z 1, It has the product representation n-s Z n=1
h(n) =
I
cpg(p,s) {
cpf(p,s)}-1
Theorem 5.1 easily follows from Theorem 2.1 for the following lemma.
Lemma 5.2. Denote by iil the set of multiplicative functions, and by Aq the intersection ,1q = .'eq n Al. Then, for any q > 1,
Aq c ;. Remark. The assertion is wrong if q = 1 (see Exercise 8). Proof of Lemma 5.2. Choose a real number E > 0 so small that (1+2E)/q is less than 1; this choice is possible since q > 1. Denote the conjugate index by q', q' = q/(q-1). HoLDER's inequality gives 21
k2
p-k
P
If(Pk)I =
Z
pk
2
If(p)1
p
1qE k
x {: Z 5 {Z Z If(pk)Iq p-(1+e)kl1/q p k=2 J ` p k22
p
k-(1 - q )
p k-(1
q
)q }1/q'
Using the assumption f e Eq and partial summation, resp. the inequality
(1 - q )
q' > 2, both of the double series on the right-hand-side of the above formula are convergent.
The assumption f e ,q implies If(n)I s Y ' nl/q with some constant y. Assuming q < 2 without loss of generality, we obtain Z P-2 If(p)I2 S Y P
Y
p-2+1/q If(p)I P
{ P
p-(1+0 If(P)Iq}1/q . { Z P P
and because f e gq , E > 0 and the right-hand side is finite, and so f
it/q" J
1, the expression on
,
e
§.
11
III 6. The Theorem of Saffari and Daboussi, II
117
111.6. THE THEOREM OF SAFFARI AND DABOUSSI, II
The theorem mentioned In the title was proved in Chapter II, 8. We prove it again easily In the special case of multiplicative functions. Theorem 6.1. Assume that A and B are subsets of IN with the property IN = A x B (direct product); a e A, b e
so every n e N Is representable as a product n
B,
In a unique way. (1)
If
I
b-1
beB
is convergent, then the density
S(A) =
lim
Z
x-1
x - oo
b-1 } 1
1 = {
acA, asx
bcB
exists.
(2) If
b--1 be
eB
Is divergent, then the density S(A) is equal to zero.
The representability of the semi-group (N,
)
as a direct product
IN = A x B is equivalent to the possibility of decomposing the constant function I as a convolution product I = a * p of the characteristic functions a, (i of the sets A, resp. B.
In the special case where a and
are multiplicative, it is easy to 1(*) deduce Theorem 6.1 (1) from Theorem 2.1. First a = * (i next the constant function I is in a", a is in , and a and I are related: 1 = 1(p) = a(p) + P(p) for primes p implies (i
1
1 - a(p) P
I
= E P-1'( 1 - a(p) P
ZP P
1
R(P)
I0-1(m)(n)I and so a = I * n-1 where < co. Theorem 4.1 and the relation y (p,l) yJ3 (p,1) = 91(p,l) give the assertion. 3-1(*),
These remarks may be considered as a hint that there might be a more general version of Theorem 2.1 in which the assumption of multiplicatlvity can be weakened.
Related Arithmetical Functions
118
111.7. APPLICATION TO ALMOST-PERIODIC FUNCTIONS
Denote by ,al [resp. B1 the C-vector-space of linear combinations of exponential sums n H ea(n) = exp ( 2ni a n ), a real [resp. a c Q], and
denote by .2 the C-vector-space of linear combinations of RAMANwAN sums
cr: n H E dl(n,r) Using the semi-norm
(r/d)
d
Iif II
a mod r, (a,r) = 1
ea/r(n).
{lim sup x 1 .1 If(n)Iq}1/q, the spaces
=
q
=
x -- >m
nsx
- closure of .al [q-almost-periodic functions],
- closure of £ [q-limit-periodic functions],
- closure of 2 [q-almost-even functions]
may be constructed. These spaces will be studied in Chapters VI and VII in more detail. In this section we are going to prove the following result. Theorem 7.1. Assume that the multiplicative arithmetical functions f and that g E and g are related, that f E (i)
(ii)
(iii) (iv)
if f Al, then g E .al if f E D then g c D if f E 31, then g E $1; if 11flll < co, then IIg111 < w. e
Remark. These assertions follow from the fact that g = f * h with a "small" function h. So the existence of such a function is also a sufficient condition for Theorem 7.1.
Proof of Theorem 7.1.
The assumptions imply g = f * h, where
(i)
Z n=1 n-1 I h(n) I < Ep (see Theorem 2.1). Given s > 0, put S = s
and choose N so large that
z n2N
n-1. Ih(n)I < s.
(1 + II f II1
)-1
111.7. Application to Almost-Periodic Functions
119
Select a finite linear combination of exponentlals near f; more exactly, f - t < e', where choose t = Z"'. ax ea with the property II
II
( 2:n=1 n-1 Ih(n)I )-1. Define the function H by
e' = s
h(n), if n s N, 0 if n > N.
H(n) =
,
The convolution e * H is in A: (eo,.
* H)(n) =
aln aSN h(d)
e,,(n/d) =
d
,,(n),
N
with the function
d a(n) =
(7.1)
10
d1'n,
if
,
e a (n/d)
,
if din.
The relation Ilsmsd exp( 2iti m n) = d if din, and = 0 otherwise, d
implies
d ,,,(n)
(7.2)
and
da
so
=
d-1
lsmsd
,
exp 2iti m d
'` +
d
J
n ),
a is a linear combination of exponentials, c A, and eC * H is in A. Using the inequality end
in
fact
11F * G1i1 5 IIGIII' En 1 n-1' IF(n)I
(7.3)
(see Exercise 6), we find IIg - t*HIII 5 11(f-t)*HII1 + IIf*(h-H)II1 5
II f - t ii
5 e'
E n=1
E n_1 n-'- IH(n)I +
II f II1
IIf 111 s
< 2e.
EN n 1 Ih(n)I
Any of the (finitely many) functions e * H is in A; therefore t * H _ 21a ea*H ) is in A, and g is in A1. (ii) If a E Q, then cad a is in £, as shown above, and so ea*H is in D
if a is rational. These remarks are sufficient to obtain a proof of (ii) by repeating the proof of (I) almost verbatim.
Related Arithmetical Functions
120
is In 2; the reason for this being that the function d(n) = cr(n/d) if din, and = 0 other(ill). The convolution cr * H : n H ZdSN
wise, is even modulo and so is a linear combination of RAMANUJAN sums. Using this observation the proof of (iii) Is performed as before.
(iv) is left as Exercise 9.
0 is multiplicative and that the series
Example. Assume that f e (7.4)
f (p) - 1
P-1
)
P
is absolutely convergent. Then f is in 21. A special case of this example is
the function n H µ2(n). Other examples are n H n
and
nH
This follows from Theorem 7.1: condition (7.4) states that f is related to the constant function 1 e §*, and this function is obviously in 2 c 21
Results for membership to $2 are not as smooth as the results of Theorem 7.1. We have to use a norm-estimate similar to estimate (7.3) used in the proof of Theorem 7.1.
Lemma 7.2. Let F and G be arithmetical functions, where F has finite (see (5.1)). Assume that G = F * h, where h satisfies the condition n-'
Z n-1
(7.5)
'
Ih(n)I < oo.
Then IIGII2 < oo. More precisely,
IIGII2 s ( zn
(7.6)
n1
'
Ih(n)I)
IIFII2
Proof. II F*h2II2 = lim sup N-1'
nN s lim sup N-1 2 nsN
s lim sup N
N-1
21
I h * F(n) 12
An
dsN tsN
I h(d) F(n/d)I
NO F(n/t)i to & nsN,
I
n-o mod [d,t]
Using the CAUCHY-SCHWARZ inequality, we obtain
III.7. Application to Almost-Periodic Functions
II F-h 1122 5 lim sup
Z
121
Z
N - oo dsN tsN
nsN
I
F(n/d)12
n-O(d)
x (N/t)-t E nsN
I F(n/t) 12
)t/2
n=O (t)
and (7.6) is proven.
Now, using the same ideas as for the proof of Theorem 7.1, we immediately obtain the following theorem.
Theorem 7.3. Suppose that f is an arithmetical function In B2, and h: IN
--) C satisfies condition (7.5). Then, again,
g = f*h is in
the function
.2
111.8. EXERCISES
1) Let f be a multiplicative function in 0*. For a fixed integer m define g(n) = f(n) if g.c.d.(n,m) = 1, and g(n) = 0 otherwise.
Prove: if f has a mean-value M(f), then the mean-value M(g) exists and is equal to M(g) = M(f)
II (i P-k
f(pk) )-1.
pim k=O
2) Assume that f is a strongly multiplicative arithmetical function, for
which the mean-value M(f) exists and is non-zero. Let g be the arithmetical function defined in Exercise 1). Prove: the mean-value M(g) exists and
M(g) = M(f)PIm n PP
1
rr (1 +
f(p) - 1
PIm
)-1
P
3) Consider the additive function A = II - w. For the densities defined in section 4, use Theorem 4.S to obtain the formulae so = rr ( 1- p-2) = 6 1 at, then F(pt") = F( pro'')'
1stsT
Proof. (i) follows from F(q) = F(q 0 = F(q) + F($); (i) implies (ii); follows from Z F( pta,) = F(q) = 1stsT
F(p1P-OC,
F(Ptot,
q) _ 2125tST
)
+ F(p1a).
(iv) If n = IT poe, then, using (ii) and (iii),
F(n) = 2]plq F( pP°) = Zplq F(
pmin(PP'cx,))
= F(gcd(n,q)).
O
Generalization. If F Is q-periodic and e-nearly-additive (this means: if gcd(n,m) = 1, then F(n) - F(m)1 < s ), then (i') (ill)
if gcd(C,q) = 1, If R Z at, then F(pto) - F(pt ' )I < 2s.
IF(8)I
vcHLET character X mod N with the following properties:
If pIN and k e IN, then f(pk) = 0. (ii) If pIN, then the function k H is constant and $ 0. (iii) There are at most finitely many primes p for which I for some exponent k. (i)
rv,2. Simple Properties
133
IV.2. SIMPLE PROPERTIES
First we prove the following theorem. Theorem 2.1. The algebras Bu, Du, Au are BANACH algebras (and so are complete with respect to . IIu), and the supremum-norm has the properties (i) - (v) and (o) of section IV. 1. II
Proof. Let us prove, for example, (iii) for Bu: given some s > 0, there are functions F, G in B satisfying
E Bu and f - F IIU < s, II g - G IIu < E. Then f + g - (F + G ) 11U < s, and so (f + g) E $u. Next, Bu is an algebra: given f, g in Bu , and s > 0, there are functions F, G In B satisfying II f - F IIu < s, II g - G II u < E. Then II F G - f g IIu f, g
II
II
5 IIf - F
GIILL
0, there is, by the WEIER-
STRASS Theorem, a polynomial P(X) with real coefficients, satisfying IP(x)- lxi < e in -M s x s M. A being an algebra, the function P(p) Is in A, and ll P(w) - Iwl Ilu < E, and so 191 a Au, Therefore, Ifl Is in Au. The formulae max (f,g) = 2(f +g)+ 2''If-gl, min(f,g) = 1(f + g) show the assertions concerning max(f,g) and min(f,g).
If 9 is in D, resp. A, then the shifted function Ya is clearly in £, resp. A (similarly, cpb,a is in £, resp..); and 9. is near fa if 9 is near f. Theorem 2.3. If f is in Au then the mean-value M(f) exists. Moreover, the FOURIER coefficients
?(a) = and the RAMANUJAN coefficients
ar(f) = {p(r))-1 . exist.
Proof. Without loss of generality, let f e > 0, there exists a function
FE
e
Au be real-valued. Given
.A with the property
F(n) - e < f(n) < F(n) + e
for every n e N. The mean-value M(F) exists, therefore the difference
of the upper and lower mean-value of f, is 1)
Of course, p Is (see, for example, CORDLINEANLI 11968] ) almost-peri-
odic, and so there are c-translation numbers for cp; these are also c-translation numbers for 191, and so jp1 Is In Au.
Iy.2. Simple Properties
135
IM_(f) - M _(f)1 s e,
and so M(f) exists. If f E A", then
are also in A", and
and
thus the assertions about the FouRIBR and RAMANWAN coefficients are clear.
Theorem 2.4. Let f E A", and let X c C be a compact set with the following property: there is some S > 0 such that UN B(f(n), S) c X. B(f(n), S) denotes the ball with radius S around f(n). Assume that : .2' --3 C is LIPSCHITZ-con tin uous;
so there is a constant L with the property 14(z) - 4(z')I s L
z - z' I, if z, z' E.X.
I
Then the composed function N - C
4rf
is again in A". The same result is valid In .$".
Proof. Let a be less than 8. If F in A is near f, f - F II" < e, then the values of f and F are in X; by the LIPSCHITZ-continuity, III0f - 40F11" s L e. We have to show that °F is in A". According to the complex 11
version of the WEIERSTRASS Approximation Theorem, there is a polynomial P(z,z) with complex coefficients, so that
14(z) - P(z,z) Thus
4(F(n)) - P(F(n),F(n))
I
I
8,
where S > 0, and If there Is an angle
E C, Iarg(z) -a1 > 8) free of values of f, then log(f) is in A".
Theorem 2.4 is a special case of the next, more general, theorem.
Theorem 2.6. Let f c A" (resp. f c $"), and, for y > 0,
Uniformly Almost-Periodic Arithmetical Functions
136
KY = { z E C: In E IN with the property If(n) - zI < y }. Then, for every continuous function 4): KY - C, the composed function i{r°f
:
IN -) C
is again in Au (resp. in 2u). Proof. The function f is bounded, therefore the closure KY/2 is compact and 4), restricted to KY/2, is uniformly continuous. Given s > 0, there is a 8, 0 < 8 < Zy such that I4)(z) - c)(z')I < s for all z, z' E KY/2' 1z-z'1 < S.
Choose a function F in A (resp. in .) near f, If - F IIu < S. Then Il4of-40FilusE.
If f E Su, F E $, then 40F E 2, and 4rof E $u. If f E Au, F
then 94u by the WEIERSTRAss Approximation Theorem (as in the c .F proof of Theorem 2.4). Therefore, 4rof is in Au. E
,v4,
The next result contains a characterization of the additive functions of to Su. Theorem 2.7.
(1)
If f is in Au and Is additive, then scup
(1)
If f is in 21 u, then lim
(ii)
k -> m
I
f(pk)
< oo.
P
f(pk) exists for every prime.
(2) If f is additive and If relations (1) and (li) are true, then f is In Su.
(3) If f is in Du and is additive, then (ii) is true, (4) Therefore, the Intersection of the vector-space of additive functions with Du is equal to the intersection of this space with Bu. Proof. (l.i) Without loss of generality, f is real-valued; f is uniformly bounded, and so IZ f(pk)I s Ilfllu, summed over any finite set of primepowers for which f(pk) z 0 (and the same is true for every finite set of prime-powers for which f(pk) < 0 ). These remarks imply
IV.2 Simple Properties
137
21 sup I f(pk)I s 2 p
k
IIfIIu+ 1.
(l.ii) The values f(pk) are bounded, so there is a subsequence k1 < k2 < Ln-1, if kr z K1(n). , for which f(pk,) is convergent, I
Choose Fn a . 7 1 near f,
1 1F
n - f IIu < n-if k 2 K2(n) is large, then the
values Fn(pk) are constant, and thus I L - f(Pk)I s I L - f(pk,) I
+
f(pk,) - Fn(Pk,)I + IF n( Pk) - f (Pk)I
0. There are constants P 0 and k0 (depending on E), so that P
Po kP
If(Pk)I < E, and If(pk) - g(P)I
0, then Proof. Du and $u are algebras, and so (1) is clear for Du and 5u. Approx-
imating f by a finite linear combination of functions e./r' it is easy to reduce assertion (2) to the problem of showing that n H e./r( P(n) ) is in D'; but, due to P(n+r) = P(n) mod r this function is periodic and so
itisin2.. Finally, we give the following uniqueness theorem. Theorem 2.9. Assume that q z 1, f E Du and II f II
=
{ lim sup X-'- E X -> co
nsx
II f II
If(n) Iq
q
= 0, where
}l/q.
Then f = 0.
Proof. Assume, on the contrary, that there is some no, for which If(no)I = 8 > 0. Choose E = there is a function F in £ near f, so that II f-F Ilu m in N)
co
fn(t) = f(t) exists for every t
e
IR,
(ii) the limit f is continuous at 0.
Then f is the characteristic function of F. For a proof see, for example, LUKACS [1970]. The application to arithmetical functions rests on the fact that
xHFN(x)=N-l
nsN, g(n)sx
1
is a distribution function, if g is a real-valued arithmetical function; x H FN(x) is non-decreasing, FN(-co) = 0, FN(co) = 1, and is right-
IV.3. Limit Distributions
141
continuous. Its characteristic function is
eitx dFN()
fN(t) = f N
nsN
eitx d( nsN,
('
N-1
=
gW ()sx
1
l /
e it-g(n)
Theorem 3.1. Let g e Au be real-valued. Then there is a limit-distribution
F (x) (if x is a point of continuity for F
F(x) = lim
N>m N
for g.
The proof is a direct application of the continuity theorem (note the fact that it is not assumed that g is additive). If t is an arbitrary real number, then the function n H exp( itg(n) ) is in Au according to Corollary 2.5. Thus the sequence of characteristic functions fN(t) =
N-1 -
2:nsN exp( itg(n)
)
converges for n - co to the mean-value
M( n H exp(itg(n)))
NO.
The inequality eiu
-l
I
=
I
i fo e'
fu
di; I
S
I
e1E I
I
di; I
s K Jul,
if u is in the disc B(O,R), with some constant K = K(R) [if u is real or, more generally, if Im(u) 2 0, then it is possible to take K = 1], gives N1
.
nsN
( exp( itg(n) ) - 1
s
N-1
E
nsN
and so, as N tends to infinity, If(t) - f(0)I s
so that f is continuous at t = 0. An application of the continuity theorem for characteristic functions gives the assertion. The theorem given above may be extended to classes of arithmetical functions that are much larger than Au; this will be done in Chapter VI, 8 A.
Uniformly Almost-Periodic Arithmetical Functions
142
IV.4. GELFAND's THEORY: MAXIMAL IDEAL SPACES
Some notions and definitions from functional analysis are used In this section. We refer to the Appendix, A.6. The algebras Bu Bu c u are commutative BANACH algebras with identity element e = 1, and there is the "standard" involution f H f (complex conjugation) satisfying f f Ilu = f IIu2. So these spaces are II
II
B -algebras, and, according to GELFAND and NAIMARK's Theorem, these
algebras are essentially algebras of continuous functions on the [compact] maximal ideal space A. The GELFAND transform f
(4.1) 'Bu
-a e(AS)
f
resp. ^:
:
- C, f (h) = h(f) ), u --) L°(AV) resp. ^: Au -, L°(A.4)
is an isometric isomorphism in each case.
IV.4.A. The maximal ideal space A$ of 8'. a) Construction of some algebra-homomorphisms. Clearly, for any integer n e IN, the evaluations hn : f H f(n) are elements
of Ate. Next, for any prime p, and for f e Su, the limit f (pm) = liimm f(pk)
exists, as shown in Theorem 2.7, and so the functions hPm : f H f(pm) are elements of AJ9. More generally, given exponents kp, 0 s kp s co, a (complex) value f(X) can be defined for the vector
X - (kp)p prime in the following manner 2): consider the increasing sequence nr of positive integers 2)
We think of the sequence of primes being ordered according to size. An Integer n may be described as a special vector X, where at most finitely many of the kv are non-zero and none Is Infinity.
N.4. Gelfand's Theory: Maximal Ideal Spaces
T nr =Ispsr 11
min(r,k pp
)
Pp
,
143
r = 1,2, ...,
with the property nrlnr+l . Then f(X) = lim
r-4 m
f(n
r
exists3), and
hx: f H f (X)
Is an element of A$. All these functions hx are different, as can be seen by evaluating hx on suitable RAMANLI,JAN sums cqC, where q is prime.
Our goal is to prove that we obtained all the elements of A$. Before doing this, we calculate the values of hx at RAMANLJJAN sums cge for prime powers qz Obviously (giving the greatest common divisor on the right-hand-side a natural interpretation), .
hx(cge) = cqE( gcd(f pkP, qe )), and this equals
cge(q') = y(q'), cgt(g8-1) = - qt-1
(4.2) =
0,
if kq z e, if
kq = E-1,
if kq 1, Since h(f) a spec(f), and spec(cge) is {p(q), -1 } if t = 1, and (1) if $ = 0, there are at most three possibilities for choosing the value h(cq ). However, not every choice is admissible. The relations {
(4.3')
Cpm'Cpe = ep(p")'cpm
,
if m > $,
and
cpZ'cp,e = p(P')'(c1 + cp +... + cppe_1) + (pe-2pe-1).cpt
(4.3")
imply (using the fact that h is an algebra-homomorphism; q denotes a prime) (a)
h(cgm) = 0, if h(cgg) = 0 and m > $,
h(cgt) * 0 and 0 s m < l;,
(b) h(cgm) = cp(gm), if
(c) h(cgt) < 0 is possible for at most one 2
(d) if
( q fixed
h(cg.,j) = 0 but h(cge) $ 0, then h(cgt) _ - qe-1 < 0
Therefore, either h(cgm) = p(qm) for any m 2 0 (define kq = case), or there exists an exponent kq such that
y(gf'), (4.4)
h(cC) =
co
.
in that
if tskq,
- qR-1, if t = kq+ 1, 0, if k > kq+ 1.
Then, for the vector X = (kq)q prime , we obtain h = ham, and so A2 is completely determined.
c) Topology. The GELFAND topology of A GELFAND transform (4.1) n
f: 0
is the weakest topology that makes every n
f(h) = h(f) continuous. So, for any prime power qz and any open set 0 in C, the sets cqg
h e A; h(cge) e 0 }
IV.4. Gelfand's Theory: Maximal Ideal Spaces
14S
are open. Therefore, using (4.4), the sets
kp arbitrary for p $ q, kq 2 Z 1,
( hx. where Z e IN, and
( hx
kp arbitrary for p $ q, kq = Z-1 }
are open. Choosing these sets as a subbasis for the topology, we see that every f is continuous. For: Given s > 0 and f, choose g = 2:1SreR satisfying IIf-gllu < ze. Assume that h e As , h = h_T, X = (kp(h)), is given. An open neighbourhood U(h) of h is defined by the condition h* a U(h) iff h*= hx,*, and kp(h*) = kP(h) for any psR.
Then h(g) = h*(g) for any h* in U(h), and so I
f (h) - f (h*)I
h(f) - h*(f) S
li
f-g
11
I
s
I
h(f) - h(g) I+
I
h*(f) - h*(g)I
+IIf - g 11u 0, then there ncN exists a [real-valued] square-root g of f in 8
u.
e) Applications.
The following result is well-known and can also be derived from the WEIERSTRASS approximation theorem (see Corollary 2.5); we deduce it from our knowledge of A2.
Corollary 4.4. Assume that f
E R u. Then i / f
E S u if and only If
infnENlf(n)I Is positive.
Proof. If 1/f e $u , then this function is bounded and so Ifl is bounded from below. On the other hand, according to GELFAND's Theory (see RUDIN [1966], 18.17), i/f E 8u if for any h E A2 the value h(f) is not zero. The values h(f) are given as certain limits in section 2, and the condition Ifl z S obviously implies that all these limits are non-zero, and corollary 4.4 is proved. This corollary may be extended considerably.
Theorem 4.S. Let f c ,$u be given. If the function F is holomorphic in some region of C, Including the range f (A$) off , then the com-
posed function F,f is in L'(&) and thus is equal to some g gE
,8 u. Therefore, Fof Is in Bu again.
Except for the last sentence, this is a specialization of L. H. LooMis [1953], 24 D. Next, g = Fof implies h(g) = F(h(f)) for any h in A., and so the assertion is true if F is a polynomial [then F(h(f)) = h(F(f)]. The general case follows from this. In the case of multiplicative functions, the following results are true.
Theorem 4.6a. Let f E Ru be given. If f is multiplicative, then f(pk) = 0 Is possible for at most finitely many primes p, and the same argument gives the following stronger version of Theorem 4.6a.
Theorem 4.6b. Let f E $u be given. If S > 0 and f is multiplicative,
then there are at most finitely many primes with the property
N.4. Gelfand's Theory: Maximal Ideal Spaces
f (p k) -
11
147
> S For some k.
proof. f (hxo) = 1 where X o = (kp), kp = 0 for any p. Given E = 28, then there is some neighbourhood 14 of h with the property I f (h) - I I < E for h in R. . But this neighbourhood contains all ham, with kp arbitrary
except for finitely many primes; for these exceptional primes kp = 0 may be taken. Next, f being multiplicative,
?(h) = lim JI f(pmin(kp,L)) L-> ao psL
and this implies, by a suitable choice of the kp , and noting If (h) - 1 I < E, that If(pk )- II > E is impossible for any "non-exceptional" prime and any k. 11
IV.4.B. The maximal ideal space A.
a) Embedding of A
of D°
in rciN IT
Define, using the abbreviation Wr = exp(2ni/r), an element fre £ by fr(n) = Wr. The set of functions (4.7)
{ f* ,
1 s t s r, gcd(t,r) = 1, r = 1,2,... }
is a basis of D. A function f in D is r-periodic for some r, and so 1/f is again r-periodic and in D c DLL, if f does not assume the value zero. Therefore, spec(fr)
=
{Wr, 1 s j s r }.
If h e 0s,, then (4.8)
h(fr)= Wr(r,h)
where j(r,h) is some uniquely determined integer modulo r, depending on h. Thus we obtain a map (4.9)
cp: 0 -3 IT
defined by p(h) = ( J(r,h) )r=12
rcN
, where h and j are related by (4.8).
Obviously, cp is infective.
Examples. (1) If f is a periodic function with period M, and if H is a homomorphism in Ate, then H(f) = f(j(M,H)).
Proof. H(el/r) = er(j(r,H)). The FouRiER expansion f =
a F+eµ/M
Uniformly Almost-Periodic Arithmetical Functions
148
implies the result. (2) If g is in Du, and G is M-periodic, IIg - G 11u< e, then I H(g) - g(j (M,H)) I < 2e for every H in A... (This depends on the fact that JH(f)h s 1117114.)
(3) If hr is the evaluation homomorphism f H f(r), then j (k,h,) = r mod k for k = 1, 2, ... .
A
b) The Priifer Ring 7L
For any n e IN consider the residue class ring 7L//n-7L with discrete topology. If mmn, then there is a continuous projection (4.10)
7tm,n
a mod n ) y ( a mod m ).
7L/n.7L
The set X = IT Z/r.7 with the product topology is a compact HAUSrciN
DORFF space, and the set (4.11)
7L = { (an) E X , an E 7L/n.7 and ltm,n((Xn) = am, if mmn }
is a closed subspace of X and therefore is again compact (and HAUSDORFF). Note that IN is dense in 7L; the reason is that, given an element (ar )r in 7L, and given positive integers ri .., rN there exists an integer m c IN satisfying m = ari mod ri for I s I s N. ,
,
Since fr-s = fr It follows that j(rs,h) ° j(r,h) mod r for any h
e
A2,.
Therefore, the image of the map cp is contained in Z. A
c) Surjectivity of p: A2, -3 7L
Let some element ((xr)r in 7L be given. Our aim is to construct an algebra-homomorphism h A2, satisfying cp(h) = (ar )r. Define a linear map h: 2) -4 C on the elements of the basis of 2) by E
h(fr) = car, - °cr,
1 s k s r, gcd(k,r) = 1, r
and extend h linearly to B. Then h is multiplicative on 2): assume first that gcd(r,s) = 1; then the relation r-l-as = ( s-k +
mod rs
implies
h(f' fs) = h(fr) h(fs ). This is also true If gcd(r,s) $ 1; without loss of generality, r and s may
IV.4. Gelfand's Theory: Maximal Ideal Spaces
149
be assumed to be powers of the same prime, and then the assertion is easily checked. v h is continuous on £: given an element $ e B, 4) _ 11,,1N av-frk there exists an m e IN, for which m = (Xrv mod rv for 1 s v s N . Since
h(4)) = 4)(m), we obtain s I4)(m)I 5
and so h is continuous on D. This space being dense in Vu, h may be extended continuously to cp(h) = (ar)r=1,2,...
an
algebra-homomorphism
of
Bu,
and
'
d) Continuity of p : AZ) -> 7. . Fix ak E
I
s k s N, with the property an = am mod m if min.
Then V(a1, ..., aN)
A 7L
, ak= ak for 1 s k s N
}
is a typical basis element of the [product] topology of Z. Moreover, he cp-1(V(a1 ..., aN )) if and only if h(fk wk k for any k in 1 s k s N. This is equivalent to f k(h) = wk k, 1 s k s N, where f k is the GELFAND transform of fk.
If Llk is a neighbourhood of wkk not containing any other kth root of unity, then it follows that 1
(V(a1
N ^ -1 ..., aN )) = kn1 fk (uk
is an open set in the GELFAND topology of A.., and so 9 is continuous. A Since A., and 7L are compact Hausdorff spaces, 9 is a homeomorphism. Thus we obtain the following theorem.
Theorem 4.7. The maximal space A., Is homeomorphic with the PrUfer Ring Z, defined In (4.11).
Remark 1. The evaluation homomorphisms hn are dense in A.). Proof of Remark 1. Given H in AM, choose a neigbourhood u(H) "defined
by R"; this means that h e u(H) iff j(r,h) = j(r,H) for r in Define the integer n as j(R!, H). Then (4.12)
n = j(r,H) mod r for r = 1,2,...,R,
1
s r s R.
Uniformly Almost-Periodic Arithmetical Functions
ISO
and hn is obviously in U(H).
11
e) Arithmetical Applications
Next, we apply our knowledge of the maximal ideal space to the problem
of the characterization of additive and multiplicative functions in $u. Some of the results have already been proved In section 2 using ad hoc elementary methods from number theory. In (1943] N. G. DE BRUIN characterized multiplicative, almost-periodic arithmetical functions. Additive, almost-periodic functions were characterized by E. R. VAN KAMPEN (1940). The results are as follows.
Theorem 4.8. Assume f to be fibre-constant. Then f Is In 2 u if and only if limk , . f(pk) exists for every prime p. This result Is not true for Z)u, as the example of a character X satisfying X(p) * 0, 1 shows.
Remark 2. f is termed fibre-constant if there Is a prime q such that f(n) = f(gcd(n,q°°)) for any n. Obviously, limk--> f(pk) exists for any prime p * q trivially.
Theorem 4.9. An additive function Is In Bu if and only if (4.13)
and
lim k-o
exists for any prime
f(pk)
E sup I f(pk) I < 0
(4.14)
k
p
Theorem 4.10. A multiplicative function Is In 8u holds and If (4.IS)
E sup f(pk) p
k
I
if
and only if (4.13)
I I< OD
is true.
Remark 3. If f is in Bu then the GELFAND transform f is continuous at h., where X = (kP )P , and kq = 00, kP = 0, If p * q. All the func-
tions h.., where kP = kP = 0 for p * q, and kq = L, L sufficiently large, are near hx, , and thus the limit relation (4.13) is true.
N.4. Gelfand's Theory: Maximal Ideal Spaces
151
The proof of Theorem 4.8 now follows from the preceding remark and
the fact that for fibre-constant functions f(h) may be defined in an obvious manner using the limit relation (4.13) at q. The resulting function f is obviously continuous and so f is in $u .
For additive functions in Du we prove the following theorem. Theorem 4.11. IF f Is in Du and additive, then limk
-4 . f(pk) exists for
every prime p, and relation (4.14) is true; therefore an additive function from Du is in fact already In ,$u. proof. Given s
>
0, choose an M-periodic function F in £ satisfying
II f - F Ilu < ; E. Then F is E - nearly additive, and so, according to section 1, IF(pO)I
k-1.
Clearly, this function C(pk,H), defined on arguments H in As, Is an And the extension of cP"the values C(pk,hn) being equal to k function H H C(p ,H) is continuous since the sets
d={HEAs k p ]} are open in A. So C(pk,
is the GELFAND transform of cP.. Using the multiplicativity of the RAMANUJAN sums with respect to the index, we obtain the transforms of all RAMANL[JAN sums cr. .
)
The mean-value M: Su -, C, f H M(f) is a non-negative (that is, f z 0 implies M(f) z 0 ) linear functional on Ru. Due to the obvious relation IM(f)I s Ilfllu it is continuous. The map (6.3)
M": F H M(L(F)), M": e(AB) -) C,
Uniformly Almost-Periodic Arithmetical Functions
158
is nothing more than an extension of the mean-value-functional M to L°(A2), and so Mu: t'(A ) -) C
a non-negative linear functional; it is continuous (I M"( F)I 5 IIFII ). Then Rtasz's Theorem (see Appendix A.3) immediately gives the following result. is
Theorem 6.1. There exists a complete and regular probability measure µ, defined on a o-algebra 4, containing the Bore] sets of h$ , with
the property fA F dµ = M"(F) = M(L(F)).
(6.4)
for every F E L°(02). So the mean-value M(f) = limX
can be represented as an integral,
_ x 1 nSX f(n) of functions f in
M(f) = fA f
(6.5)
In fact, it will be proved
h2 = 11 {
(6.6)
dµ.
that µ is a product measure. Write 1,
P = 11 IN,
p, pZ, ..., p°° }
and define probability measures µ
P
on the factors IN by P
µP(Pk) = p
(6.7)
1u
(iP(p
Then µP is defined on the Borel sets sets of IN ). The product measure
)
= 0.
of INP (
these are all sub-
P
(6.8)
11 µP
is defined on the least a-algebra f = 11 with the property that P P all the projections iP: A .i -* IN are f-78(N )-measurable ( this means P P that 7CP-1(AP)
E P for any Borel set AP in 78(N
P
Proposition 6.2. The product o-algebra JP = 11 P a-algebra of Borel sets in A A.
)
).
is equal to the
IV.6 Integration of Uniformly Almost-Even Functions
159
proof. Both the c-algebras mentioned in the proposition are generated by the measurable rectangles TT 7Z p, where 7Zp C NP and 7Z.p = NP for all but a finite number of primes p. This is true for .' by definition of the product (j-algebra; and by definition of the topology of A. it Is
clear that all the measurable rectangles IT 7.p are Borel sets, and that all these rectangles belong to the Borel sets. Example 3. Denote by 50 a finite set of primes, and, with each p e 9), associate an integer (including co) m(p), 0 s m(p) s oo. Characterize an element h in A. by the vector { kp(h)}p of "exponents". The set
Y = { h e 02: kp(h) = m(p) for each p in
9)
}
has measure p m(p).Zp
µ(Y) = fA XY dµ =
where tp = 1 - p-1, if m(p) < co, $p = I otherwise. The expression p-' is to be interpreted as zero.
Proof. (a) Let m(p) < co for each p in P. Y is open and closed, so the characteristic function XY is continuous, and µ(Y) =
lim
N-1
N -4c,
nsN m(p) In
The relation n e Y is equivalent to p iff gcd(a,b) = using Zdl (a,b)µ(d) = 1
m(pp )= mp and P = p `
1,
XY(n).
for each p in P. Therefore,
and writing P _
{
P1' "' ' pr}'
p r`r'', we obtain
,
21 N-' ' n5N X r Y(n) = d p µ(d) ...dp, µ(d) 1
N-1
1. N n=0 mod (Pd,...d,)
For N --) co, this expression tends to IT p-m(p).( I _ p-1 ). PET
(b) In the case where m(p) = co for at least one contained in every set
Zm={heAs,m(p)Zm} with measure
P in
the set Y is
Uniformly Almost-Periodic Arithmetical Functions
160
i1(Zm) = (1- p-1 ). (p-m + P-m-1 + ...) = p-m,
according to case (a), and thus µ(Y) = 0.
11
Example 4. The set N of positive integers (embedded in 02) N c A2 has measure zero. Enumerating the primes as p1 < p2 < ..., the measure of the set
Yr,s = { x ( As, mp(x) = 0 for pr S p s ps } is
r IT s
(
1 - pp
)
(according to example 3). Therefore,
Yr = (l Y r,s skr has measure IT ( I - p-1) = 0 and the assertion follows from IN c U Yr r
P>P,
coincides with the pro-
Theorem 6.3. The measure space (As, duct measure space
(TTNp,P,µ ). P
Proof. According to HEWITT-STROMBERG [196S] a product measure is
determined uniquely by the values of the measure on measurable rectangles. Without loss of generality, these may be taken as TT A p, where
AP = NP for p
z
po, and AP =
{pm(P)
;
m(p)
a
M(p)},
where
M(p) c {0} U N U {co}. Then
µ (TT Ap) = Y1p
IL (Ap) =
On the other hand, the same expression is obtained for µ(TT A ) P
example 3.
Corollary 6.4. If 9 Is a finite non-void set of primes, and f(p): NP -4 is µ P-Integrable for each prime p e 9', then the function f : A$ - C, h H TT f(P)(n (h)) pcT
P
is [L-measurable, and fo.Y3
f dµ
pTT C 71
fQom] dµp. O. P
Of course, iLp is the projection of 0$ to its "p-th factor" N .
by
p
C
IV.6. Integration of Uniformly Almost-Even Functions
161
Example S. The continuous extension of the RAMANWAN sums cr to "
AR was given mean-value is
(see (6.2)) as Cr
M(cr) = f
0R
=
.
x
H TTp"Ilr c' (r()P). Therefore, the
cr" dµ = IT fIN' c"P` dµ P P`IIr
Pk-1,(1-p 1).p-(k-1) + p(pk) .
-
pr
= 0 If r z 2, and l If r =
EmZk P
m'(1-p ') }
1.
Similarly, 2m(r).
7 (1-p), if r Z 2, pr
M(IcrI) =
if r =
1,
1,
and
f
mdilp = Pk-' (p _1) if k = m, otherwise = 0, P
and, therefore, the orthogonality relations M(cr'cs) = p(r) if r=s, and = 0 otherwise, are proved again. The final example 6 gives a calculation of the RAMANUJAN coefficients
for functions f in Bu which are finite products of fibre-constant functions f(P),
f = IT f(P), where 91 is a finite set of primes. pcT
equals f f cU dµ, and, this being a product
The mean-value over simpler integrals,
(1-p-1)( f(1) + p-1.f(P)+p 2f(p2)+...)
M (f'cr) _ TI
Pei),PXr
x TT
k PE9'.P II r
(1 - p-1).{-f(pk-1)+PP(Pk)'Emikp-'.f(pm)}
if all primes dividing r are in 91, and otherwise r
0.
An extension of the integral to the larger class (vector-space!) ,
q(
3 ( A2), µ), where q 2 1,
Uniformly Almost-Periodic Arithmetical Functions
162
of measurable functions F: 0B -4 C with the property fo., IFIq dµ < co is possible. Identifying functions F, G with IIF-GIIq :_ =
ffo.IF-GIq dp 11/q = 0,
the well-known L'-spaces are obtained. L2 is a complex HILBERT-space with inner product
F,G > = ro$
dµ.
The set of functions r = 1,2, ... is an orthonormal basis in L2. This follows from the fact that the continuous functions on A are dense in L2 and the linear combinations of [extensions of] RAMANUJAN sums are dense in L°(A2).
Finally, we note that a more powerful theory of integration of arithmetical functions was developed by E. V. NovosELov about 1962-1964, and the most powerful theory of integration, due to J.-L. MAUCLAIRE, is presented in his monograph [1986].
IV.7. EXERCISES
1) The pointwise product of an r-even and a t-even function is { l.c.m[r, t]}-
even. Prove this and a similar result for periodic functions.
2) Let r e N and f the indicator-function 1rN of the set r W. Calculate the RAMANLUAN coefficients a (f) and the FoURIER series of f. d
3) For given r e N, calculate the RAMANLUAN coefficients ad(f) for the function f defined by f(n) = if gcd(n,r) = 1 and f(n) = 0, if 1
gcd(n,r) >
1.
IV.8. Exercises
163
Solution: p (d)
` $
,
if dir,
e
a (f)
d
4)
Prove:
= 0,
if d Xr.
the quotient space £/$ is of infinite dimension.
(Hint: the residue-classes e1,r + .2, r = 3, 4, ... are pairwise different.) 5)
6)
The quotient space AID is of infinite dimension.
Let k be a positive Integer, and f an arithmetical function. Put fk(n)
= f(gcd(n,k)). Prove the equivalence of the following three
properties: (1) f e 2 ",
for every s > 0 there is a k in IN so that f-fk 11u - E. (3) the set { fk, k e N } is relatively compact in the set of bounded (2)
II
functions with the topology induced by II. Ii
.
7) Prove: the assumptions f e A, inf.,, If(n)I s 2, do not imply f-1 a A. (Hint: f(n) = 1 + 2 e,.(n).) 8) Let f e Au have no zeros. If If1-1 a A", prove that f-1 a '4u. 9) Give a formula for the GELFAND transform Cr a 0 NL(JAN sum cr a 2".
of the RAMA-
10) Describe a countable base for the system of neighbourhoods of the evaluation homomorphism h1 a A2.
11) Let {n1} be a sequence of positive integers with the property that the least prime-divisor pI of nl tends to infinity. Then the evaluation homomorphisms hn converge in A2 to h1. r
12) Let {n1) be an increasing sequence of positive integers with the fol-
lowing property: for every R e N there exists an io e N so that n) = nl mod r for 1 s r s R, j a I> i o. Then, in A.., the evaluation homomorphisms h n are convergent. [Example: nl = j!]
13) Prove: the evaluation homomorphisms hn, n = subset of A9).
1,
2, ...,
are a dense
Uniformly Almost-Periodic Arithmetical Functions
164
14) Show in detail that A. is homeomorphic to
= I {1, p, p2,
iN*
... ,
P,),
P
where each factor is the ALEXANDROFF-one- point-compactification
of the discrete space 0, p, p2,
... }.
1S) Let Xr s be the characteristic function of the residue-class s mod r. Mr
Prove: `Xr,s', otherwise.
l
Xr,s ) = 0 if (rt,r2)] Is1-s2I, and = {lcm[rt,r2]}
t
a
16) (A. HILDEBRAND).
a) Prove, for all Integers qt, q2, N, the asymptotic formula
N-t'Zn,N
q,(n)-cg1(n)
c
= Sq
b) There exists a positive constant ct such that the inequality N-t' 2:nsN I zgSQ a q' cq(n)12 s ct ZgSQ IagI2. cp(q) Is
true for all integers N, Q s N' and all complex sequences
(at, a2, ..., aQ).
c) Prove, by dualizing this inequality, qSQ
Here,
I cn.(n)I = n $ 0,
and thus the system (1.4) is solvable.
Theorem 1.1 is not very interesting, because the coefficients br are not the "natural ones". Convergence of the RAMANUJAN expansion (1.2) for a large class of functions was proved by A. HILDEBRAND [1984].
Theorem 1.2. If f is an arithmetical function in
$u
then the RAMA-
NLUAN expansion a c (k) = f(k) r
15rN {Mn}-1 s
QN-2 '
even functions fn = ErSQ partial sums at the point k,
satisfying IIfnIIu = 1, with "large"
I ErsW a sequence of even functions, {Mn}-1
FN = E nsN
fn.
this sequence is a II.IIu-CAUCHYsequence with limit F In Su. Then IIF - FN IIu s En,N s QN-2 Our goal is to show that the RAMANUJAN expansion of F is divergent For IIfnIIu = 1 and E
{Mn)-
u
1
\
The integral from 1 to infinity is equal to n=1 n-1 gk(n) log(n), which can be evaluated in the usual manner, replacing log n by Zp.lln log(pk) and inverting the order of summation. This calculation is a little laborious and is left as Exercise 3. The result is
f1
n-t, gk(ndu
u-
n2 u
{
`Pkk'
P
{P(Plog(P) +
p k
p l log(p)}.
This formula concludes the proof of Lemma 3.1.
11
The proof of Theorem 1.5 rests on estimates of the following incomplete sums over the MOBIUS function: M(n,z) = Zdln,dsz µ(d) and
M1(n,z) _
(3.3)
din,dsz µ(d) log(z/d) = f u-1 M(n,u)du.
Lemma 3.2. Uniformly in z z
I
II n H M1(n,z) Ill m
N-1
I
M(n,z)
I
log(2x)
( log(2x) \ 218 \ log(2z) )
where p1(n) Is the least prime factor of n, pnln(1) = co, and where (3.4)
S=I-
2)/log(2) = 0.0860713...
The more difficult result is the second one; Lemma 3.2 can be deduced from Lemma 3.3 in the following way. First, for pklln and u z 1, there
Ramanujan Expansions of Uniformly Almost-Even Functions
174
is an identity M(n,u) =
(3.S)
and so (3.6)
M1(n,z) =
f zz/ p
M(p
U-1
du.
For the proof of Lemma 3.2 we have to estimate the sum EnsN IM1(n,z)I.
We split this sum EnsN IM1(n,z)I according to the condition Pmtn(n) > z [resp. s z] and use (3.6) in the second sum ( with p = pntn(n), (Pmin(n))klln, n' = n/(PmIn(n))k N-1
E
nz E + N-1 .
IM (n,z)I
E
nsN,p ,(n)sz
1
('z
J z/p
u-t
IM(n',u)I du.
Ordering according to p = pmin(n) s z , we obtain, after replacing n' by n, N-1
E IM (n,z)I s
nsN +
E
1
Ek
t
IM (n,z)I 1
p-k ,
zip
In the first sum, according to the condition pmin(n) > z and the definition of M1(n,z) there is only one divisor d of n with d s z, namely d = 1, so in this sum M1(n,z) = log z. The sum EnsN,p(n)>z I equals m. EnsN,gcd(n,k)=1 1, where k =P:Kz IT p, and this sum is (3.8)
EnsN,gcd(n,k)=1 1 = Edlk µ(d)
'
(d + e(d)) = N
2k l
+ R,
where Ie(d)I s 1, and IRI s t(k). So, for N - co, the first sum on the right- hand side of (3.7) approaches lim N --> m
N-1
E nsN,p_(n)>z
log z =psz rl (
1-p-1)
log z x
Pm,,(n)>x
So, for the proof of Lemma 3.3 it is sufficient to deduce the following two lemmas:
Lemma 3.4. Uniformly In z Z x 2 lim
N --* ao
N-1
Z
1
IM(n,z)12
x Lemma 3.5. Uniformly in z Z x Z
Nlim
N -1
Z
1
nsN, M(n,z)*O
1
« { log 2x }
1
log 2x log
2z
where 8 was defined in Lemma 3.3.
The proof of these two lemmas is given in section S.
V.4. PROOF OF THEOREM 1.5
In order to prove Theorem 1.5 (and thus the other theorems of this chapter) first we have to transform the sum defining the kernel SQ k(n), using cr(k) = Yk' Igcd(r,k) k' µ(r/k ' ) We obtain S
Qk(n) = Zk,
Ik
k'' Z
= Zk' Ik k'
2]
rsQ k' Ir
rsQ/k' (p(rk'
cr(n) [L(r/k' ) )}-1
Crk. (n) µ(r)
If r is squarefree, then factorize r = r' r", where r' Ik' and gcd(r",k') = 1, in order to obtain
Ramanujan Expansions of Uniformly Almost-Even Functions
176
SQ k(n) _
k'
k' Ik
','k' (n)'i1.(r') cp(r' k')
r' sQ/k'
r"sQ/k' r'
r'ik
p(r")
gcd(r",k')=1
Using the inequality Icr(n)I s p(r) and the abbreviation
k' (n) = 1rsz,gcd(r,k')=1 (cp(r))-1 where z > 0, the estimate (4.1)
TZ
µ(r)' Cr(n),
rXlk' ITQ/k' r' k, (n) I
s k Ik
k
s Ek' Ik
k
ISQ k(n) I
'
follows, and thus (4.2)
IISQ,k
II
I
Er' Ik'
IITQ/k,
r' ,k' III.
For (4.2) it suffices to show the estimate supy'I IITZ kill < ao
(4.3)
Replacing cr(n) by the usual sum over divisors of gcd(r,n) and inverting
the order of summation, we begin with
TZ k(n) = Edln d =
ErSz,dlr,gcd(r,k)=1
E
E
{p(d)}-1
r' sz/d
dln,dsz
S,µ(r)'i!(r/d)
{cp(r)}
{cp(r'
gcd(r',dk)=l
gcd(d,k)=1
The inner sum is known from Lemma 3.1. Inserting the result, we arrive at
(n) = w(k)
T
z,k
k
F(1) (n)
z,k
+
+
w(k) k
w(k) k
rC+h(k)) l J
F(2)(n) z,k
F(3)(n) z,k
(k) F(4)(n)), `
with the abbreviations FZ1k(n) = Edln,dsz,(d,k)=1 µ(d) Iog(z/d), FZ (2 )(n) =
Edln,dsz,(d,k)=l p(d),
FZ (3) (n) = Edln,dsz,(d,k)=1 µ(d) h(d),
and
z,k
1
V,4, proof of Theorem 1.5
177
FZ4k(n) = z-'. ZdIn,dsz,(d,k)=1
{p(d)}-1
Thus, in order to prove Theorem 1.5, it is sufficient to prove sup
II
F ('k
11
The treatment of the [semi-] norm N1
,
i = 1, 2, 3, 4.
II
F(4)111 is easy: we estimate z,k
< co
1
E IFZ k(n)I = N-'-z-'
{p(d)}-1
,
d3/2 4(d)
dsz
nsN
1
nsN/d
(d,k)=1
5
{p(d)}-1
.
Zd5z
.
d' 4(d).
N) gives
Partial summation (beginning with %sN the estimate 0(1) for the last expression, uniformly in z. {p(n)}-1
Next we show that, without loss of generality, one may assume that k = I.
for short, where Xk For the remainder of section 4, we write µk = is the characteristic function of the set of integers which are coprime with k. µk is multiplicative, and has a representation as a convolution µk = µ*hk, where hk is multiplicative, and hk(pm) = I if p1k, and zero Using this notaotherwise. The series X°°n=1 k (n) equals tion,
FZk(n) = Zdln,dsz Vk(d) - Idln,dsz d.,d"=d 21d..ln,d"sz hk(d") 2: d'I(n/d"),d' sz/d" µ(d') hk(d")
_
1(n/d"), Fz/d"
whence IIF(2)II z,k I s
dsz
k
(d)
II
F(2) z/d,I 1 11
s {p(k)j-1 k sup .a
F(2)111.
11
w,1
Similarly, a corresponding result is true when the upper index (2) replaced by (1). For the upper index (3), a careful calculation gives
FZ3k(n) = Zdln dsz,(d,k)=1 ii(d)
'
Zpld
P-1, log(p)
is
178
Ramanujan Expansions of Uniformly Almost-Even Functions
p-1.1og(P)
ZPln,psz,p1k 2:Pln,psz,pYk
'
2:dln,dsz,d=O(p),(d,k)=1 V(d)
1
log(P)
p
FZ/Plkp(n/P),
.
and a short calculation gives II
FZ3k II1 S
psz prk P-2'log(P)
Fz/p,kp II1
log(p)
k
5
II
9W
N?
P
(2)
IIFW
1
II1.
So, finally, the assertion of Theorem 1.5 is reduced to the problem of a uniform estimation for the following "Incomplete" sums over the MoBIus function: M(n,z) = dl dSZ
µ(d)
(2) = FZ (n) 11
and
M1(n,z) =
dI aSz µ(d)
log(z/d) = f 1 u-1 M(n,u)du [ = F(n) ].
Thus Theorem S follows from Lemma 3.2 and Lemma 3.3 (see section 3).
V.S. PROOF OF LEMMAS 3.4 AND 3.S
The proof of Theorem 1.5 will be finished as soon as we have proved Lemmas 3.4 and 3.5. For this purpose we need the following result on the MoBIUS function:
Uniformly in x z
1,
t 2 1, and d e IN the estimate
Enst,(n,d)=1 n i µ(n) x, and where
V.S, proof of Lemmas 3.4 and 3. S
179
L (1 4(d) = Edlk d-''µ2(d) = Ipk
+p
).
TI p, the absolute value of the sum Using the notation P(x) =psx p Y_
nst,(n,d)=1 n
1
il(n)
is equal to IE nst,(n,d)=1 s
n-1'µ(n)
'
Emlgcd(n,P(x)) µ(m)I m-1.11 2(m).Iznst/m,(n,dm)=1 n
mIP(x),mst
log-2(2u) in u >-I, which is a little Using the estimate Znsu stronger than the prime number theorem, the inner sum in (5.3), with slightly changed notation (u = t/m, dm = k), is equal to IEnsu,(n,k)=1
n-1µ(n)I
n-1. µ(n)
= IF1dsu,dlk°D d-1,
-/-u, the last expression is d-1
'4 q/JY(Aq), IT Ac(f) = f + lY(Aq).
The quotient norm is defined by (see, for example, RuDIN [1966], 18.15) IIn(f)IIq = inf {
II
f+g
11
q,
g E Jr } =
II f Ilq.
Then Aq, Dq, Bq are BANACH-spaces.
In Chapter IV, Theorem 2.9, a uniqueness theorem was proved for functions in Du. In ,A41, a theorem of this kind is not true. However, arithmetical properties such as additivity or multiplicativity have consequences on the uniqueness of functions In A1. As examples, we prove the following theorems. Theorem I.S. (Uniqueness theorem for additive functions). Assume that
f and g are additive functions In A1. If
II f - g II1
= 0, then f = g
identically.
Theorem 1.6. (Uniqueness theorem for multiplicative functions). Assume f and g are multiplicative, and both are in £1, Ilflll * 0, Ek>1 p-k If(pk)l < m for every prime p, and f-g is in the nullspace JY(41). Then f = g identically.
Remark 1. The assumption IIfII1 * 0 in Theorem 1.6 is necessary. The functions f = E, g(n) = 1 if n = 2k for some k, and g(n) = 0 otherwise, are both multiplicative and satisfy IIf - gill = 0, but f * g. Remark 2. The finiteness of the norm Ilfll for some q > 1 implies (see g III, Lemma 5.2) Ekk1 p k f(k)1 < oo for every prime p. So this condition can be omitted In Theorem 1.6 If f E Dg for some q > 1 is assu-
med. We shall see later (VII, Theorem 5.1) that this condition is also superfluous in the case q = 1.
Almost-Periodic and Almost-Even Arithmetical Function,
196
Remark 3. If f E D1 is multiplicative and non-negative, then the condition MP(f) < M(f) implies Ik21 P k. I f(pk) I < co (see Exercise 4 ).
Proof of Theorem I.S. Put h = f - g, and let pk be a fixed prime power. For any Integer N, we obtain the lower estimate ZnsN Ih(n)I
2:nsN,P`Iln Ih(n)I = JmsN/P`,Pkm Ih(pk) + h(m)I
2t
h( Pk)I
2
- G msN/p`,p4m ' 1-
msN/p Ih(m)I.
Dividing by N, for N -) co, the inequality IIhIII 2 Ih(Pk)I
.
. (1_p-1) _ p-k.11h1I1
P-k
is obtained, and the assumption IIhJi1 = 0 implies h(pk) = 0.
11
Proof of Theorem 1.6. Assume that there is an integer n0 for which f(n0) $ g(n0 ); then there
is
a prime-power pk with f(pk) $ g(pk)
Then, for every N, Z If(n)-g(n)I 2 nsN msN/p1,gcd(m,p)=1
I f(pk) - g(pk)I. E
I
I
If(m)I - Ig(pk)I
msN/p` p .I' m
.
E If(m)-g(m)I. msN/p`
Add the term I
f(pk) - g(Pk)I
.
msN/p°,PI*n
If(m)I,
to both sides of this inequality and divide the resulting inequality by N ( pk (N/pk) ). Letting N tend to infinity, we obtain, using the abbreviation MP (g) = lim"-> - x 1. 1nsx,p1n g(n) [for the existence of this mean-value, see Example 1 following Theorem 1.3], II
f -gill + If(pk) - g(Pk)I P-k .
MP(Ifi)
z If(pk) - g(Pk)I P-k
M(Ifi) - Ig(pk)I p-k
II
f_ g III,
and therefore M (Ifi) Z M(If)I, a strange result certainly, which comes P from the assumption f $ g, which is to be refuted. Next
VI.2.
properties of q-Almost-Periodic Functions
GnsN,Pln
197
If(n)I = 21 k21 XnsN,P"iIn If(n)I,
= Zkz11f(Pk)I ( msN/p" If(m)I - ZmsN/P`,PIm If(m)I), and so
Zk2O If(Pk)I "msN/p`,pIm If(m)I = Ek21If(pk)I
' Z n N/p If(m)I
Dividing by N and using the dominated convergence theorem (this is p_k If(pk)1 < oo ), we obtain possible for y = 1 k21 (1+Y) MP(IfI) = Y' M(IfI),
therefore, using the estimate MP(IfI) z M(IfI), proved above, Y (1+Y)-1 - MOM ;, MOM,
which contradicts the assumption 11f111 = M(Ifl) > 0.
VI.2. SOME PROPERTIES OF SPACES OF q-ALMOST-PERIODIC FUNCTIONS
As mentioned already in section 1, HOLDER'S inequality II1 s IIfIIq
II
IIgIIq
, ,
where q-1+q'- I = 1,
implies .v4q c s4r c '41 whenever 1 s r s q land there are corresponding
results for the other spaces - see Figure VI.2]. Starting with k = 2 (which is HOLDER'S inequality), mathematical induction gives the following
Proposition 2.1. Assume that
1/q1+l/q2+...+1/qk=1,
(2.1)
where 1 < qx < oo. Then (2.2)
11
f1 ' ...
fk 111 s IIf1IIq
IIfk11q.,
Almost-Periodic and Almost Even Arithmetical Functions
198
Proposition 2.2. Assume that all the norms appearing In equation (2.3) below are finite. Then the following assertions are true:
(2.3)
(i)
If r t
(ii)
If
(iii)
II
I
1,
q-1 + q'-' = 1, then
s q s r, then IIq 5
II g
11U.
f II,q
II
II f III 5 II f 11q 5 II f IIr 5
II g 11'.q
, .
II f IIu
II f IIq.
Proof. (iii) follows from the definition of II.IIq; the other inequalities are obtained from HOLDER'S inequality.
Theorem 2.3. Assume that 1 5 q s r < co, and q-1+ q'_ I = 1. Then (1)
2u c .fir c 8q c 21, D C Du C Dr c Z) q c Z A c v4u c Ar c Aq c A1. c
.$
(2)
11q c Z)q C Aq C A1.
(3)
2u.Bq c $g,
(4)
2q- 2q
(5)
If f
E
if f if f
E
c
..
c 21,
g c Dq, D
,$g, then Re(f), Im(f) and IfI y) q, then Re(f), Im(f) and IfI 34 q, then Re(f), Im(f) and IfI
34u.Aq c 34g.
Aq.,4g E
.Sq,
E
Dq,
E
c 341.
34q.
(6) If f, g are real-valued and both are in ,$g [resp. Dq, resp. 3487, then
max(f,g) and min(f,g) are In Bq [resp. Z) q, resp. 8q].
Proof. Assertions (1) and (2) are clear. For (3), assume that f 34g, g E Au2 E > 0; choose functions G, F in A near g, f such that E
II g-G IIu < E/(II f IIq+1), II f - F IIq < E/(IIG IIu+1). Then F- G is in A and 5
IIq + IIG'(f-F)Ilq 5 IIg-GIIu'II f IIq +
f-Fllq < 2E.
34c1, g E 4g , E > 0, choose F, G E A, II f-F IIq < E/(II g IIq+I), 1Ig-GIIq. < E/(II F IIq+1). Then F G Is in A and (using HOLDER'S inequality)
(4) If f
11
E
5
11
1+
5 11 f-FIIq'IIgIIq, + 11FIIq'IIg-GIIq. < 2E.
V1.2. properties of q-Almost-Periodic Functions
199
(5) The real or imaginary part of a function in 2 [resp. 21, resp. A ] is again in B [resp. 2), resp. A ]. If F is in S or 2, then IFI is even or periodic and so, again, is in B or D. And, using the usual approximation s f-g 11 1, the assertions arguments [and the inequality Ifl - IgI Aq Implies Ifi in 41 q. But in are proved with the exception that f in if F in A, the WEIERSTRASS Approximation Theorem gives: this case II
then IFl in
Au
II
II
(by IV. Theorem 2.2), and this is sufficient for a proof
of the remaining assertion.
Assertion (6) follows from the formulae max(f,g) = 2(f+g) + 2']f-gl, min(f,g) = 2(f+g) -elf-gl.
11
Theorem 2.4.
(1) If f Is in A1, then the mean-value M(f), the FouRIER coefficients and the RAMANUJAN coefficients NO = ar(f) = {9(r))-'- M( f ' cr )
(2.S)
exist.
(2) In A2 (and so In the subspaces .82, 2)2) there is an inner product < f,g > = M( f-9_)'
(2.6)
and the CAUCHY-SCHWARZ Inequality (2.7)
I
15 11f112'IIg112
holds.
(3) If f is in
A2,
then at most denumerably many FouRIER coefficients are non-zero, and BESSEL's Inequality EpCR/ZIf (a)12 s Ilfll22
(2.8.1)
holds. If f Is In 22, then BESSEL's inequality reads (2.8.2)
211sr 2 is left as an exercise. Theorem 2.6.
(i) If g Is In At real-valued and bounded, then, for any e > 0 there exists a function t in .v4" near g, g - t IIt < s, with the addiII
tional property IItIIu S IIgIILL
(ii) If g E Al Is bounded, then g E Aq for every q 2 1. (iii) Assume that g e Al has a bounded representative and f is in Aq. Then the pointwise product is in a4q. Remark. The same results are true for the other spaces 2q and 2q. Of course, in (i), in these cases t may be taken to be in $ [resp. in D]. If g is complex-valued, in (i) it is possible to find a t satisfying IItIIu s
IIRegII2 + IIImgIILL
s 2 IIgIILL.
Corollary 2.7. If f Is In Dq, then the functions where x Is a DIR7CHLET character, Ik is the characteristic function of the set of Integers relatively prime to k or the characteristic function
of the set of integers congruent to a mod k, where gcd(a,k) = and the pointwise product
1,
are In Dq again.
Proof of Corollary 2.7. The functions x and 1k are periodic and bounded.
The function µ2 is bounded and is in $t (this is a consequence of the Relationship Theorem from Chapter III; it2 is bounded, therefore in ;, and it is related to the constant function I y°*). E
Proof of Theorem 2.6. (1) Given E > 0, choose a real-valued trigonometric polynomial t In s4 [resp. D or 21] near f, II g-t < E. Put W
w
II
t = max{ min (ta, IIgIILL), - IIgIILL}. Then IIg-tilt 5 IIg-t*Iit < E, and t is I. v4u
[resp. 9, resp. $ for the other spaces], and IItIIu S IIgIILL (ii)
is a special case of (iii).
Almost-Periodic and Almost Even Arithmetical Functions
202
0, choose t1, t2 in Au, such that Ilf-tlllq < E, IIg-t2111 112 < Eq/(1+llgllu2+ Iltl ), and lit2llu s 2- llgllu. Then an easy computation (iii) Let E
>
shows it
g-t2 IIq s { 4q-1 Ilglluq-1
IIg-t2111 }1'q.
Therefore, Ilfg-tIt2llq 5
S
S llgllu.E +
llgllu ll(f-tl)llq +
lltl Ilu const(q)
s constt (q, llgllu)
.
Ilgllul-t/q
Ilg-t21111/q
E
Since t I t 2 is in A", Theorem 2.6 is proved.
11
Theorem 2.8. If f is in B1 [resp. D1 , resp, A1] and 11 f IIq < co, q > then f is In 2r [resp. fir, resp. 4r ] for any r in 1 s r < q.
1,
Remark 1. An additional condition is needed to secure that this result is true for r = q (see section 8). Remark 2. The assertion of Theorem 2.8 is not true for r = q, as shown by the following examples.
Example 1. The function f(n) = na if n is a square, and f(n) = 0 otherwise, has norm II f llq = 0 as long as q < 2, and it is (trivially) in ,$1 All RAMANLUAN coefficients ar(f) = M(fcr)/p(r) vanish, but nsx lf(n)l2 ti 'x, and so II f 1122 = M( Ifl2) = 2. But PARSEVAI:s equation M( lfl2) = Y_ cp(r)
lar(f)12 (see section 3) is violated, and f is not in $2.
[This example is due to J.-L. MAUCLAIRE].
Similarly, the function g(n) = -/ log n if n is a prime, else g(n) = 0, has 11g 1l, = 0, all ar(g) = 0, llgll22 = 1, and PARSEVAL's equation is violated again. Example 2. (A. HILDEBRAND). Fix q > 1, and put f(n) = 2klq if n = 2k
is a power of 2, and f(n) = 0 otherwise. Then 'if IIr = 0 if I s r < q, > 0, but f is not in 21q. 11 f Ii q [The proof runs as follows: it is easy to calculate x fq(n) nsx and to show that limx -> m x-1 Ensx fq(n) does not exist (for example, 1
VI 2. properties of q-Almost-Periodic Functions
203
2k+1-1);
let x -4 oo through the sequences 2k and
therefore the mean-
value M(f q) does not exist, and so f9 is not in Al ].
proof of Theorem 2.8. Without loss of generality, let f be real-valued. Define the truncation fK of f by f(n), if If(n)I s K, K, if f(n) > K, -K, if f(n) < - K.
fK(n) =
f E $1 implies that fK E .$1, and - being bounded - the truncation fK is in $4 for every E Z 1. Define s' > 1 by q and fix s by the equas-t+s'-1 = 1. Then, using HoLDER's inequality, tion o(x) = x-1 Ensx If(n) - fK(n)Ir s x 1
SI
'
7-nSx.lf(n)I>K If(n)Ir
iq/(s' q)
nsx
,
JJ
X-1.
l
I
li/s
nsx,lf(n)I>K
Next, Kq
Znsx,lf(n)I>K I S 1nsx,lf(n)1>K If(n)I' 5 (2
II
f IIq)q ' x,
if x is large. Hence, we arrive at lim sup A(x) s II f II q/s ' (2 II f / K ) < E, q q m X if K is chosen large enough, and so f, being near fK E .$r, is in Sr. q/s
II
We state that for real-valued functions f in ,$r the truncated function
fK tends to f in 11. "r' and that, for any E > 0, (2.10)
x-
lim sup x -* o
21
nsx,lf(n)l>K If(n)Ir
the
y ).
Therefore,
I g(n) - Q(t(n)) lap s { l g(n) - {max(0,t(n))}1/a +
max(0,t(n))
Q(t(n))
p
and using (b), (a) and (c), this becomes s 2ap-1 ,{l g°`(n) - t(n)lp + I { max(0,t(n)) }1/a - Q(t(n)) lap }.
Therefore II
g - Qo t
11,X0 S E.
This is one part of the proof. By Exercise 11 (or Corollary 2.5 (3)) h e
.94k
kc W, implies hk
order to prove the other part, put Y=(
1)
( aP )-1,
a
.a41.
In
V1.2.
properties of q-Almost-Periodic Functions
Then Y
i
1, and
c
205
N. The function h = gl/Y satisfies hY a
[according to the first part of our proof] h
therefore h«PY number c4ly is an integer, and so
a
c
.A4 ap,
.A 'x'3y. The
,A4'. Therefore, go"'
a
.r41 and
the first part of the proof again gives go' E AO . For the question of the existence of a limit distribution of real-valued
functions the following result is useful, as it has already been shown for uniformly-almost-even functions. Theorem 2.11. Let q i 1.
(1) If f E $q is real-valued with values in some finite for infinite] closed interval I = [a, b], and if the function Y': I -- C Is LIPscH[Tz-continuous (so that I W (x) -'F(y)I s L- I x - yI for some constant L > 0), then the composed function 'F ° f is in 3 q again. The result remains true, if S q is replaced by £ I or .4 q. (2) If f e 1q Is complex-valued with values In some finite [or infinite] closed rectangle R, and If the function 'F: R --) C is LipscHiTz-continuous, then the composed function 'F ° f Is in 2q again. The result remains true if $q is replaced by Dq or a4 q. (3) If f In 211 [or 01 or .A41 ] is real-valued then the function nH Is In B1 [or 01 or Al ] for any real t. (4) If q z 1, f E $q, f is real-valued, and infnEN I f(n)1 = S > 0, then f
E Bq.
Proof.
(1) Let s > 0. Choose a trigonometric polynomial t in ,$ [resp. £] near f-tNIIq < e. The values of f are in I; t* is real-valued, without loss f, of generality. If the values of t* are not in the interval I = ]a, b[, replace * t by t = min { b, max(t* a) } (with an obvious interpretation, if a or b are ± a ). t is - nearer to f than t*, therefore II f - t II < E. Then II
II
II
q
q
'1' ° t is even and so in 2 [resp. periodic and so is in £], the values of f and t are in I, 'F Is LIPscHITZ-continuous, and therefore II
'F
f - 'F o t
q
= limX sup x 1 Ex IY'(f(n))-'Y(t(n))Iq n!c
s lim sup x 1. Lq x -* co
nsx
If(n)-t(n)I q s Lq s q.
In the case where f E A q and t ,A4, the function 'F ° t is in s4" by the WEIERSTRASS Theorem, and the proof works in this case, too. E
206
Almost-Periodic and Almost Even Arithmetical Functions
(2) The complex case can be reduced to the real one. Assume that R = Cal, bI] x i Cat, b2]. Then approximate Re f by an even function t
with values in Cal, b1], and Im f by an even function t2 with values in Cat, b2]. The even function t = tI + I' t2 has values in R, and IIf- t IIq s II Ref - tI IIq + II Im f - t2 IIq. The rest may be concluded as in (1).
(3) and (4) are special cases: the functions x H exp (it x ), defined on JR, -)y-1, defined in y Z 8, are LIPSCHITZ-con_ where t is any real number, and y tinuous. Thus 1/IfI E 8q, and f-I = f' IfI-2 E . by Theorem 2.6 (ill).
Examples.
(a) If f is a bounded function in A and P a polynomial with complex coefficients, then the composed function P ° f Is also in A1. This follows from Theorem 2.11, but It could also be deduced from the fact that a bounded function in AI is in A9 for every q 2 1. (b) If f E s4q satisfies b:= sup Re(f(W)) < oo, then exp(f) E Aq. The C; reason is that exp is LIPSCHITZ-continuous In the half-plane {z E
Rezsb}. (c) If f E Aq and a:= inf(Re f(N)) > 0, then log f
a44, because the principal branch of the logarithm function Is LIPSCHITZ-continuous In E
the half-plane {z e C; Re z 2 a} with L = a Remark. If P is an integer-valued polynomial with positive values, for example P(n) = n2+1, then it is a difficult task to prove that f - P Is in Al (or has a mean-value at least) if f is in some A9. The result is not known even for the function µ2, If the degree of P is greater than two.
VI.3. PARSEVAL'S EQUATION
According to section 2 of this chapter the spaces $z C D2 C ,42 are complete vector-spaces with an "inner product"
< f, g> = M(fg). This "Inner product" Is
linear in the first argument; it satisfies
VI.3 parseval's Equation f g>
207
< g, f > and < f, f > z 0, but < f, f > = 0 is possible for
functions f $ 0. Thus, the quotient-spaces modulo null-functions,
2 c D2 c A2, are HILBERT spaces. Theorem 3.1 (PARSEVAL'S equation).
(1) I f f is in
,
2, then Er°1 cp(r)
lar(f)I2 = 11f112
where the ar(f) denote the RAMANUJAN-FOURIER coefficients
ar(f)
M(f cr), r = 1,
= r)
2, ...
.
(ii) If f Is in z2, then Y_
(iii)
r=1 11sasr,gcd(a,r)=1
If f is in
I
M(f ea/r) 12 =
11 f 112
A2, then
«E;R/
12 = 11f112
I
Corollary 3.2.
(i) The set { (p(r))" Cr r = 1, 2, ... is a complete orthonormal system in 22. If f, g are In $2, then }
,
ro 1
(P(r)
(ii) The set { e./r' r =
1,
ar(f) 2, ...,
ar(g) =
1 s a s r, gcd(a,r) =
1
} is a com-
plete orthonormal system in .V2. If f , g are in D2, then Zro
1
lsasr,gcd(a,r)=1 M(f ea/r)
M(g ea/r) = M(f g ).
(iii) If f, g are In A2, then 21 «EW./a
M(g'1«) =
First Proof. The assertions of Corollary 3.2 come from the "Elementary Theory of HILBERT space", which is sketched in Appendix A.2. According
to this theory, the validity of the PARSEVAL equation is equivalent to the denseness [with respect to II. 112 ] of the sets .$, £, A of linear combinations of RAMANUJAN sums [resp. exponential functions] in $2, D2, and A2; this is true by definition of these spaces.
Almost-Periodic and Almost Even Arithmetical Functions
208
VI.4. A SECOND PROOF FOR PARSEVAL'S FORMULA
In this section we present a second proof for PARSEVAL's equation in the space $2. Some properties, perhaps of some Interest, of arithmetical functions in $2 are exhibited, and these properties are used in the proof. Let r be a positive integer, and, for k dividing r, denote by Xk the charac-
teristic function of the set Ak = { n e W: gcd(n,r) = k). Xk
is
a function in r c . (with positive mean-value), therefore , 2 for every f in $2. Consider the linear map F
r : 22
8 r'
fy k 1r
M(xk'
Xk'
This function has the properties given in the following lemma. Lemma 4.1. (1)
F (f) = f if and only if f E 2 .
(2)
If f, g E $2, then M(Fr(f) g ) = M( f Fr(g) ).
(3)
If f e 22, then Fr(f) _ k ak ck, where ak = cP(k) M(f ck).
(4)
If f E $2, g E Br' then II f - Fr(f) 112 s II f - g 112 So Fr(f) Is
(5)
a "best" approximation in $r For every f In 22, the sequence
k1r
AR(f) = Ilf-FR!(f)112' R = 1,
2, ...,
Is monotonically decreasing to zero.
Proof. (1) A function f in 2r is constant on Ak, say equal to d k, for every k dividing r. Therefore, Fr(f) = XkIr dk Xk = f. (2) By definition of Fr and the linearity of the mean-value, we obtain
M(Fr(f) g) = kr M(me) M( Xk g). This expression is symmetric in f and g, and so also is equal to
(3) By the orthogonality of the RAMANUJAN sums the coefficient ak , using (2) and (1), equals
VI.S. An Approximation for 1-Even Functions
{p(k)}-1
ak =
. M ( Fr(f) . ck) _
{9(k)}-1
209
. M(Fr(ck) f) _
{9(k)}-1
. M(ck f).
(4) Without loss of generality, we assume that f and g are real-valued. Let x > 0, klr, and define the function Gxk: IR--jlR, y Hx
1
nsx,n c Ak
( f(n)-y)2.
This function has just one stationary point Xk(n))-1
mx = (2:nsx as x --- ) co,
z
(M(Xk))-1
. M(f Xk) + 0(1)
and this point gives the absolute minimum of Gx k. Therefore,
1.
nsx,n c A. X-1.
=
. (Znsx f(n) - Xk(n)) =
(f(n) - F (f)(n) )2 = r Z
nsx,n c Ak
x-1
(f(n) - (M(X k ))-1 M(f ' Xk)) 2
nsx,n c A.
(f(n) - mx)2 + 0(1) s x-1
I
nsx,n c Ak
( f(n) - g(n))2 + o(1).
Summing over k1r, we find for x -) co IIf - Fr(f)112 s IIf- g 112. 2
(5) At first AR+1(f) s AR(f), by (4). Now, given E > 0, there exists an even function g E . "near" f, 11 f - g 112 < E. Choose an integer R, for which g is in .Y3R. Then IIf -FRI(f) 112 < E,
again by (4).
11
Now we are ready to prove PARSEVAL's equation
ZO1 p(r)
Iar(f)12 = M(If12) =IIf IIZ
For every f E 22, and for every integer R, a standard computation gives rlRl
r
k = 11f 112
2
- rIR! Ep(r)' Iari2.
By (3), the left-hand side is If - FR!(f) 112, which converges to zero by 2 (5).
Almost-Periodic and Almost Even Arithmetical Functloa8
210
VI.S. AN APPROXIMATION FOR FUNCTIONS IN .81
In the last section, the result
112-0, as R -)m, - r was proved by [elementary] HILBERT-space methods. In this section, similar result for arithmetical functions In $1 is given. IIf
u
a
Theorem 5.1 (A. HILDEBRAND). For every function f in St Rm
(5.1)
II
f-
r R1
ar(f) cr IIt = 0, .
where ar(f) = M( f c'), r = 1, 2 JAN coefficients of the function f. {cp(r))-t
,
... denote the RAMANU-
The important feature of this result is that the coefficients of the even functions approximating f are not changed when R is increased. Note that the sequence {R!}R=1,2,... may be substituted by every sequence {nR}R-1
with the property limR---). gcd(nR, r) = r for every integer r.
2
Remark. This theorem allows us to show [again] that the Mornus func6 II µ III = 7t2 > 0, does not belong g to .Y3 t It is known from prime number theory that tion µ, with
.
Md(µ) = limx
x- 1
---> m
'nsx,n=O mod d v(n) = 0
for every integer d: Therefore, (5.2) p(r) . ar(µ) = x
l'
x-1
Zx ti(n) . cr(n) =
dr
d µ(r/d) . Md(µ) = 0.
First we collect some formulae, needed for the proof of Theorem 5.1, as follows. Lemma S.2. (1) For every Integer k, r, R satisfying rIR!, if r.}' k,
0, (5.3)
1 R!
nsR!
n=O (k)
cr(n)
cp(r) k
if rlk. '
V1.5. An Approximation for 1-Even Functions
211
of the
set
= k}, where kJ R! Is supposed. ar(f) for f E 81. Then, for every kIR!, FR! = ZriR! Cr'
Put
k
by
Denote
(2) A
=
{
n E N:
characteristic
the gcd (n, R!) Xk
function
M( f Xk) = M( FR! Xk)
(5.4)
proof. (1) The RAMANUJAN sum Cr is R! - even (and so R! - periodic)
if rIR!, therefore the left-hand side of (S.3) is equal to lim
X --> oo
x-1
c (n) =
nsx,kin r
lim x
d µ(r/d)
dr
= k-1
1
x---> m
nsx,d n,kin
gcd(d,k) .Il(r/d)
dr
IT ( gcd(pZ, k) -
= k-1
1
gcd(pt-1
k)),
pEllr
and this gives the right-hand side of (5.3). (2) FR! is R! - even, and so we obtain lim x-1 X -> m nsx,kln FR!(n) = R!
E
nsR! kin
lim
x -> m
X_
1
x
L
f(m)
a (f)
r
l
c (n)
r
Cr(m) ri R! fp(r)
n R!
R!
Cr(n)
kin =
k-1
lim
x-* m
x-1
msx
f(m)
r!k
Cr(m)'
using (1) and the fact that k divides (R!). The inner sum equal to
rk cr(m) is
k, if kim, (5.S)
n(1+cP(m)+...+
CPt(m)
p"Ilk
0, if k4' m.
Therefore, the functions FR! and f have the same mean-value on the sets Mk = { n EN: kin}, if kI R!, and so also on (S.6)
K=
Mk \ GRIP"*k
MQ.
11
C-0modk
Proof of Theorem 5.1. We start by proving that for any real-valued function f in 231 and every real-valued k-even function g the estimate
Almost-Periodic and Almost Even Arithmetical Functions
212
Ilf-FR,III
(5.7)
kIR!
holds.
FRI and g are R! - even and therefore constant on every set Ak If kIR! (for the definition of Ak see Lemma 5.2 (2)). Denote the values, taken by FR! and g on Ak, by y k I resp. 8k1. Fix k, and assume that Yk z 8k. Then
Zx If(n) - FR,(n)I = Ex (f(n)- yk) + Ex ncA
ncA.
f(n) 2
( Yk- f(n))
ncA,,
f(n)
0, choose a real-valued even func-
tion g near f, Ilf -gill < s. If g is k-even, then, according to Ilf-FRI III 0, ax are integers
=1,
(x = 1,...,k),
then the function F: n H f1(bI n+ al)
...
f k ( bk n + ak)
has a mean-value.
[This is a generalization of L. LucHT's results; this author only dealt with multiplicative functions, but he obtained product formulae for the mean-values (see L. LuCHT [1979a, 1979b]). The continuity theorem for DIRICHLET series (see the Appendix) might be helpful in calculating the mean-value of the function F given in (5) in the case of multiplica-
tive functions, but some additional conditions seem to be necessary to obtain "nice" results.] (6) If f is real-valued and is in ,94q, where q 2 1, and if the image f(N) is contained in a closed Interval I c IR, and if `1: I - C is LIPSCHITZ continuous, then the composed function To f is in .4q, and so it has a mean-value.
Examples for 'Y are the functions z '- z-1, z '- exp(z), z y log(z), etc. Of course, one has to be careful about f(N), and some assumptions on the values of f are necessary before (6) or other versions of Theorem 2.11 are applicable.
VI.7. Arithmetical Applications
(a) Let q 2 1. If f then 1/ f .v4 q.
E
217
34 q is real-valued, and if infnfiN If(n)I = 8 > 0,
E
(b) If f e .1q is complex-valued, and if supnEiRe(f(n)) s K < co, then exp(f) E Aq. (c) If IF
E A q is complex-valued, and if Inf... Re (f(n)) > 8 > 0, then
log IF E Aq.
The calculation of the mean-value can (given appropriate circumstances) be dealt with by an application of the continuity theorem for DIRICHLET series:
Theorem 7.1. If f: IN -i C has a mean-value M(f), then
M(f) = lima-),
(7.1)
1+
n=t f(n)'n
In particular, if f is multiplicative, then the calculation of the limit (7.1) often is rather simple. Proof. The existence of the limit M(f) Implies
I f(n) =
o(x), as x --3 oo.
nsx
Partial summation gives, as long as d >
Ex f(n)-n-' = Ex f(n)
.
x-' - f1
1,
Yu f(n)
du,
and so
f
o(u),u-°-1
du
1
= M(f) o
(0-1)-1
+
0((o-1)-t)
as o -4 1+.
The asymptotic relation i;(o) = ((j-l) + 0((0-1) ), as o -4 1+, gives the assertion.
According to the continuity theorem for characteristic functions (see Chapter IV, section 3) the question of the existence of a limit distribution in the sense of probability theory is a problem of the existence (and continuity) of certain mean-values. We prove the following theorem.
Theorem 7.2. If g Is a real-valued arithmetical function in .s41, then there
Is a limit distribution for g; this means that the limit N-1 . n{n s N; g(n) s x lim (x)
N-m
a
218
Almost-Periodic and Almost Even Arithmetical Function,
exists In the sense of probability theory.
For the proof it has to be shown that the mean-value Mt = M(n H exp{ itg(n)})
(7.2)
exists for any real t, and that the function t H Mt is continuous at t = 0. According to Theorem 2.11, the function n H exp{ itg(n)} Is in 011; hence the mean-values Mt exist. The continuity of t H Mt follows from the estimate x-1 (eltg(n) _ I) s lim sup x 1 Z lim nsxll x -- m / x -3 M nsx ,
=
Itl
IIg111.
VI.7.B. Applications to Power-Series with Multiplicative Coefficients.
Given an arithmetical function f, the region of analyticity for the generating power-series (7.3)
F(z) = 1'=i f(n)
.
zn
may be of some interest. In some sense "most" power series with radius of convergence equal to 1 are non-continuable across the unit disc in the complex plane [see, for example, L. BIEBERBACH [19SS]]. Of course,
a number theorist would like to obtain an answer to the question of non-continuability of F(z) if the coefficients of this power series are arithmetical functions with some arithmetical property. G. POLYA and G. SZEGO's Theorems [see BIEBERBACH [1955]] state:
if the coefficients f(n) of the power series (7.3) with radius of convergence equal to one are integers [ resp. assume at most finitely many
distinct values], then either F represents a rational function or it
is
analytically non-continuable beyond the unit circle.
For multiplicative arithmetical functions L. LuCHT and F. TUTTAS [1979] proved the following result.
V1.7. Arithmetical Applications
219
If f is a multiplicative function with finite (semi-]norm II f Ill, and if the mean- value M(f) exists and is non-zero, then the power series F(z) = I '=i f(n) z° is non-continuable beyond Its circle of convergence if and only if f(p k-1)
9(p )
$
vak
for Infinitely many prime-powers pk. Otherwise F(z) represents a rational function.
This theorem relates special properties of the coefficients of the power series to the global behaviour of the function represented by this series. and Z'=1 For example, the power series Y -'=i are non-continuable. The LUCHT-TUTTAS condition is, in fact, a condition related with the RAMANLUAN coefficients of the arithmetical function f. We are going to show that the property "Multlpllcativlty" does not play an essential role; more important is that the RAMANLUAN coefficients ar(f) _ {p(r))-1 M(f c) do not vanish "too often". Theorem 7.3. Let f e 82.
(i) If Infinitely many of the RAMANUTAN-(FoURIERJ-coefficients ar(f)
{(p(r)}-1 M(f
Cr) are non-zero, then F(z) _ I 'f(n)-z' is
non-continuable beyond the unit circle.
(ii) If only finitely many coefficients ar(f) are non-zero, and if f Is represented (pointwisel by its RAMANUTAN expansion, f(n)
ar(f)
cr(n), for n = 1,
2, ...,
then the power-series F(z) = -n=t f(n) zn represents a rational function.
Remark 1. By HILDBBRAND's Theorem (V, Theorem 1.2) the RAMANLUAN
expansion is convergent to the correct values f(n) if f is in 2u. Later we shall show that the same is true for multiplicative functions in 212, supposed that M(f) $ 0 (see VIII, Theorem 5.1). Remark 2. Using formulae for the RAMANLUAN coefficients, which will
be deduced in Chapter VIII (see VIII, Theorem 4.4), it is easy to show that in case of multiplicative functions the non-vanishing condition of
Almost-Periodic and Almost Even Arithmetical Functiop$
220
infinitely many RAMANLUAN coefficients is equivalent to LUCHT's condition given above.
Remark 3. The assumption f e $2 may be replaced by f e 21. The proof has to be changed in so far as PARSEVAi s equation has to be replaced by a result by A. HILDEBRAND, proved in section S (Theorem S.O.
Remark 4. Differentiation does not destroy the property of being rational
or non-continuable beyond the unit circle. Therefore, the result can easily be extended by replacing the assumption f e $2 with:
There is some non-negative integer k such that n H Example. Theorem 7.3 is no longer true if f f
e Ou. This may be seen from the function
f,
n-k. f(n) is in 22.
$2 is replaced by
e
defined by the uni-
formly convergent series f(n) = G1sk<m
This function is in Lu; the power series n=1
f(n) zn =
Zk
2
exp (2ni/k z )
1 - exp( 2iti/k ) z } -1
k=1
is continuable beyond the unit circle, but it is not a rational function. Proof of Theorem 7.3. PARSEVAL's equation gives (7.4)
II
f - lisrsR ar(f)' Cr
1122 = Er>R (p(r)' Jar(f)12 < E
if R z R.(E) is sufficiently large. The generating power series for the function G1srsR
is R(Z) _
n=1 1 2:1srsR ar(f)'cr(n) }
= 2: 1srsR ar(f)
Zm
. z"
mod r WZ .
(1-mz)-1,
where m runs through the primitive rth roots of unity, m =mgr =
gcd(a,r) = 1.
This function R(z) is a rational function and is "near" the following sense: IZ
n=1
f(n)'zn - 9{(z)
I
< 2E
' (
1-IzI
)-1
n=1
f(n)-z' in
VI.7. Arithmetical Applications
221
if Jzi < 1 is near 1. This may be seen from (7.4), using partial summation.
Therefore, if z = t. ma r, 0 < t < 1, t -) 1-, and if ar(f) $ 0, then I
2:-n=1
f(n)-z'
and so mar is a singular point for F(z). But if ar(f) 4 0 infinitely often,
then the corresponding points mar are dense on the unit circle, and the non-continuability of F(z) is proved. [The asserted denseness of the points mar may be deduced from a Theorem from CH. HOOLEY [Acta Arithm. 8, 1963], given as follows. Theorem 7.4 (CH. HOOLEY). Denote by I = al < a2 0, (8.1)
fE
2I
we see that Z nsx
(f'q(n) - f2r(n))2 = Z f' (n) + fr(n) - 2 Z fi (q+r)(n). nsx nsx nsx
By Theorem 2.8, f is in ,fir; DABOUSSI's Theorem gives fr E 2 ), and so
the mean-value M(fr) = M( f (q+r)
f Ilr
exists. The same argument applies to
Therefore, II f,
- f'rll22= IIfIiqq + IIfoorr - 2
IIf1I5cq+r)
(q+r)'
Making use of (C) we obtain lim
r-> q-
Ilf3q _ frllZ = 0.
f is in Sr, therefore f'r E S2 (again by Theorem 2.9). Being approximated
by functions in 22, the function f'q itself is in ,$2. Using Theorem 2.9 once more, the function f is in Sq.
229
VI.9. Exercises
VI.9. EXERCISES
1) Give a [simple] direct proof for the fact that arithmetical functions in A 1 have a mean-value.
2) If f : IN -4 IR is an integer-valued function in $", then f is in $. Give an integer-valued function in 21 which is not in B.
3) Denote by ADD resp. ADDS the set of additive [resp. strongly additive] functions. Prove that these are subspaces of CIN, and that the
II1 - completion of (ADD n,1) [resp. of (ADDS n,l) ] is a subspace of ,1. II
4) Assume that f e D1 is a non-negative multiplicative arithmetical function. Denote by MP (f) the limit lima -> x-1 nsx Pin f(n). Prove that for every prime Zk21 P_k f(pk) < co if and only if
MP (f) $ M(f). 5) Let f be a multiplicative function in 01. For every prime power p k prove
lima
(a)
-
m
x-1. 1nsx,P`IIn f(n) = p-k. f(pk) (M(f) - MP (f) ), x-1
pf(p,) ( M(f)- MP(f)), Gnsx,Pn f(n) = Iekk if the series on the right-hand side converges absolutely.
(b)
lima
6) Prove Theorem 2.11 (3) directly.
7) Let y > 0 be an irrational number. Denote by g(n) the number of positive Integers m with the property Cy m ] = n. Prove:
(a) g is in A2 (b) Put S = y-1 - [y-11. Then the FOURIER coefficients of the func-
tion g are g (a) = y-1, if a = 0,
g(a)21ciaY)
1
(e
and g(a) = 0 otherwise.
27cI ocy (8-1)_
-1
7L,a$0,
Almost-Even and Almost-Periodic Arithmetical Functiops
230
(c) What does PARSEVAL's equation mean? Answer:
X'=1
I
n-2. sln2(rt8n) =
,7E2
(S - S2 ), where 0 s 8 < 1.
8) Give a proof for PARSEVAL's equation in ,Z2, using methods similar
to those used in section 4. Hint: Ak = {n E IN; n = k mod r},
9) If f is in
S1
Fr(f) =
and S > 0, then the function h, h(n) =
f (n) If(n)If 8-1
Jf(n)j > 8,
f(n), if If(n)Is 8,
belongs to ,$1. .V1
and every residue-class s mod r the mean-value limx -i M X-1 nsx, n a s mod r f(n) exists. Prove this result for coprime r, s, using the formula
10) For every function f E
lnsx, n - s mod r f(n) = {P(r)}-1 ZX mod r (X(s)' Ensx X(n) f(n) ). 11) If q1 > 1, ..., qk > 1, q1-1 + ... + qk-1 =1, and f1 E .44+,
then prove that the product f1
f2
...
fk E
4 k,
fk Is in A1.
12) Let klr, where k and r are positive integers. Calculate the meanvalue of the indicator-function of the set {neN; gcd(n, r) = k}.
Photographs of Mathematicians
231
A
E. WIRSING
r H. DABOUSSI
R. RANICIN
I
P. D. T. A. ELLLIOTT
H. DELANGE
A. RENYI (1921-1970)
232
Photographs of Mathematicians {
Jk
$
A. SELBERG
I
1
M. JUTILA & M. N. HUXLEY
H. E. RICHERT
A. KARACUBA
C. L. SIEGEL
A. Ivic
(1896-1981)
J.-L. MAUCLAIRE
M. NAIR
233
Chapter VII
The Theorems of ELLIOTT and DABoussi
ABSTRACT. This chapter deals with multiplicative arithmetical functions
f, and relations between the values of these functions taken at prime powers, and the almost periodic behaviour of f. More exactly, we prove that the convergence of four series, summing the values of f at primes, respectively prime powers [with appropriate weights], implies that f is in 2q, and (If in addition the mean-value M(f) is supposed to be non-zero) vice versa. For this part of the proof we use an approach due to H. DELANGE and H. DABOUSSi 119762 in the special case where q = 2; the general case is reduced to this special case using the properties of spaces of almost-periodic functions obtained In Chapter VI. Finally, DABOUSS7's characterization of multiplicative functions in ,A4q with non-empty spectrum is deduced.
The Theorems of Elliott and Daboussi
234
V I I.1. INTRODUCTION
As shown in the preceding chapter, q-almost-even and q-almost-periodic
functions have nice and interesting properties; for example, there are mean-value results for these functions (see VI.7) results concerning
the existence of limit distributions and some results on the global behaviour of power series with almost-even coefficients. These results seem to provide sufficient motivation in the search for a, hopefully,
rather simple characterization of functions belonging to the spaces s44 D Z) q D $4 of almost-periodic functions, defined in VI.1. Of course, in number theory we look for functions having some distinguishing arithmetical properties, and the most common of these properties are additivity and multiplicativity. According to the heuristics outlined in Chapter 111.1, conditions character-
izing membership of an arithmetical function to, say, X34, ought to be formulated using the values of f at primes and prime powers.
Historically, theorems of this kind were given for the first time in connection with the problem of the characterization of multiplicative functions with a non-zero mean-value. The E. WIRSING Theorem, proved
in 11.4, is an example of the fact that assumptions about the behaviour
in the mean of values of a multiplicative function, taken at primes, imply asymptotic formulae for the sum Z f(n). But these results do ns x not characterize multiplicative functions with a non-zero mean-value. In 1961, H. DELANGE proved the following theorem.
Theorem 1.1. Let f: N -) C be a multiplicative function satisfying IfI s 1. Then the following conditions are equivalent: (1.1)
The mean-value M(f) = lim x-I Z f(n) exists and is non-zero. X --) nsx (i)
The series S1(f)
(1.2)
=
21
p-I (f(p) - 1) is convergent,
P
(ii)
Osk< m
p
0 for all primes p.
Introduction
23S
Remark. The assumption IfI s 1 Implies that
2 P kf(pk)I for every prime p 2 3. Therefore, as did DELANGE, the validity of (1.211) is to be assumed only for p = 2, and it may be substituted by IEosk 5/4
If(P)Iq
are convergent,
(iii) the series
S3,q (f) =pEkk2 E P-k If(pk)Iq is convergent.
Remarks. 1) The series SI(f) is conditionally convergent, the primes being ordered canonically according to their size. The other series are 'absolutely convergent.
2) In the special case where q = 2, condition (ii) is equivalent to the convergence of the series (ii')
S2(f) = E P-1 P
.
I f(p) - 1 I2.
The Theorems of Elliott and Daboussi
236
Using this notation, P. D. T. A. ELLIOTT [197S] proved the following theorem.
Theorem 1.3. Assume that f : N -* C is a multiplicative function, and assume that q > 1. Then the following conditions are equivalent. and the mean-value M(f) exists and is non-zero.
< oo
(1.3)
II f II
(1.4)
f Is in 9 q and condition (1.211) is satisfied,
In
9
this chapter we are going to show that the convergence of the
series in Definition 1.2 implies, in fact, that the multiplicative function f is in 8q (Theorem 4.1). Furthermore, following DABOUSSI and DELAN-
GE, we prove (Theorem 5.1) that for any multiplicative function f with mean-value M(f) $ 0 the following properties are equivalent:
Finally, we characterize multiplicative functions in Aq, possessing a non-void spectrum (see Theorem 6.1). We begin with some rather simple consequences of the condition If
II f II
9
II f 11q< oo.
< oo, then there exists some positive constant c such that If(n)I s c nt"q for every n E IN,
(1.5)
and [by partial summation from nSx If(n)I s C x ]
1 n_t n-' If(n)Iq < oo, if Re s > Lemma 1.4. I f
I I f1 1
q
< oo for some q > 1, then, E p
X Z
1.
I
f(p) z p
< co, and
p-k. lf(pk)Ir < co for every r In I s r < q.
p k22 In particular, using the notation of Chapter III, Section 1, a multiplicative arithmetical function f, satisfying II f 11q < oo, belongs to the
set
III I.1. Introduction
237
{f:IN - C, f multiplicative, Z I
f(P) I2< oo, 2:
p-k
1:
If(pk)I < m}.
p kz2
p
P
proof. Choose an e > 0 such that I + 2 E < q. HOLDER'S inequality and (1.5)
imply P
x
I ff(p) 12 5
p
I (n)k
c.
I
p2 - (2+e)/q
Psx S C .(
-LAY)j p(1+e)
Psx
\1/q (
q' \1/q 1 { Psx l p2 (2+e)/q j J
By (1.6) and the choice of s, both series on the right converge for x - -co. Similarly, with s > 0, 1+2E < q the estimate ,
p-k(1+e),If(Pk)Iglr/q.(2i z p-k(1-E a_ )`1 q
E
p kz2
p kz2 psx
/
P cx
p k22 p"sx
proves the convergence of the second series. Example. The following example shows that an extension of Lemma 1.4 to r = q is not possible. Define a multiplicative function f by f(pk) = 0
if p > 2 or k Is odd, and f(2k) = (t-1 , 22e)1/q if k = 2 Then I I f IIq = 0, but X 21
p kz2
l;
is even.
p-k. If(Pk) I q = 21 2-k , fq(2k) kz2
Lemma I.S. Let q > 1, f: W - C be multiplicative,
II f II
g
= Do.
< oo, and assume
that the mean-value M(f) exists and is non-zero. Then there exists a prime p 0 with the properties M(P)(f) =x limes
(1)
x-1 ns n
f(n) = M(f)'
{pf(p,1)}-
for every prime p z p0, and (2)
M
(d)
(f) =
x lim -. -
x-1
nsx,(n,d)=1
f(n) = M(f) j7{ W (p l) Pld
}-1
f
for every positive integer d which consists only of primes p Z po.
(3) M (f) = P
lim x-1 x -+ -
nsx, n=O mod p
f(n) = M(f) {pf(p,1)-1} {(pf(P,1)}-1
Remark. If f is 2-multiplicative, so that f(pk) = 0 for every k Z 2, then
The Theorems of Elliott and Daboussl
238
the mean-values in question are given by
M(d)(f) = M(f) II
M(P) (f) = M(f) { 1 + P-1' f(p) }-1,
P
fppl
MP ( f ) = M
Proof. In Re s 2
1,
{I+
P-1. f(p)
Id
{ 1+p-1 f(p)
(1.5) implies
Ik
p-ks , f(pk) 15 c
{ pt - 1/q - 1} 1,
1
therefore there is some p0 such that [recalling the abbreviation pf(P,s) = 1 + P-'-f(p) + for every prime p 2 po, pf(p,s)
in Re s z
C. 1.
{ pi - 1/q - 1
}-1
z 1 - C.
{ PD
1/q -
1
1
1}
2z
Let p* be a fixed prime greater than or equal to p0,
Define a multiplicative function g by
f(Pk), if p * p",
if p = p
0,
.
functions f and g are related, f is in ;, IgI s Ifl, therefore g e §. For every prime p Z p0 the factor
0, as x -3 co. Therefore, r°°a-t, (a(s) - a( s)) dt. a(s-1) P-1-s' (f(p) - 1 ) = J0 (2.1) l zpsxp-
P
Having proved that the two limits lim
(2.2)
s -0+
a(9) - a (9) dt = 0,
a-t
o
and Xp-1-s
lim s --> 0+ P
(2.3)
(f(p)-I)=a
exist, relation (2.1) gives the existence of lims _+ 0+ a(s-1), so that the series S1(f) is convergent.
For a proof of (2.2) we apply LEBESGUE's Dominated Convergence Theorem. In order to be able to do so, we have to estimate the difference a(9) - a (s) by an integrable function of t, uniformly in s. In 0 < y < z, the CAUCHY-SCHWARZ inequality yields la(z) -
a(y)I2 = exp(y) < p P-1
P
p-1
exp(z)
If(P) - 1 I2/
(f(P) - 1)
(
exp(y)
exp(1/s)
p-1. 1
f(p) - 1 12)
log t
The Theorems of Elliott and Daboussl
242
tends to zero as s -) 0+. LEBESGUE's Dominated Convergence Theorem gives assertion (2.2).
For (2.3), the existence of the mean-value M(f) implies En=1 n-3 -f(n) ti s-1 )-1 for s --- 1+ by partial summation (see VI, Theorem 7.1), and so [as s - 1+ ] Nn) I + f(o)-1 + f(p')-f(p) + ... M(f). p ( p ) = -1(s) Y--= P 1
In particular, no one of the factors
(l +
f(1'
+
p.
f(p')-f(p) pa.
+ ) is
zero. The product over the primes is split into a finite product II(...),
IT
f(p)-1
1 +
the product IT ( p>L
P
1 + f(o)-1
p:aL
)
/ and the product
1-1
1
P
p> L
+ F(P)-1 p.
+
f(P')- f(P) Pa.
+
... ).
If L is chosen large enough [so that I p-1 (f(p)-1) I s 2 ], then the last product is absolutely convergent in Re(s) z 1. Therefore lim
r(1+ F(P)-1 )=p$0
s - 1+ p>L
P
exists. Taking logarithms and using the absolute convergence of the series
P
{log(1+
f(p)-1
p,
)-
f(p)-1
Iin Re sZ1,
exists. Thus (2.3) is true. Z f(p)-1 p. s 4 1+ p> L
one sees that lim
3) For the convergence of S3(f) _ assuming 1 < s s 2, with (2.4)
_'(s)
IT n=1
p Xk=2
P- k I f(pk)I2,
_ P-s) (1 +
one starts,
p-ks.lf(Pk)12 k=1
P
The finiteness of II f 112 implies the boundedness of the left-hand side
< s s 2; hence any partial product of the right-hand side is s c1, p-s say. Let f(p) = min { I f(p) I, 4 }. Then + f*(p)2 s c21, where in
I
1
c2 = 52 . We use
1+ x z exp( x- x2) in x 2- 2 For every factor of (2.4) and for every K z 2 we obtain
K 2If(Pk)I2 1
p-s \1 +k
1
If pks)I2 ) >
\1
P-s) (I -
+
f pP) 2)
(1
+ c2 k
)
VII.2. Multiplicative Functions in 8'
a
243
K If(Pk)I2 1+ c 2 . k_2 pks
)
exp (
f *(p)2- 1- f'"(p)4+1 p
p2.
S
I.
Using IT (1+ x ) 2 Y_ xp for x z 0, and letting s tend to 1+, we obtain P
P 29Y
lf(pkk)12
E
psY k=2
P
P &Y
C,
c
1
exp 1
p
f"(p)2+1
\ PSY
p
-1 PSY
f'"(p)2- 1) p
for every y 2 2 and K 2 2. The series on the right-hand side are dominated by P p-2 = 0(1), resp. by If(p) -1
If(p)IsS/4
1
+
=
If(p)I>S/4 P
P
Therefore, the partial sums of
S1 (f) + S1(f) + 0(S2(f)) = n(1). elf(pk)l
p
k
are bounded and S32(f) is
convergent. This concludes the proof of Proposition 2.1.
VII.3. CRITERIA FOR MULTIPLICATIVE FUNCTIONS TO BELONG TO 81
In this section we give another partial answer to the problem of characterizing multiplicative functions in 2q. We show that the condition f E 9q implies that f is in 2 , and II f II < oo. First a rather special result is proved. q
Lemma 3.1. Assume that f: IP - C, and, for every prime p, If(p) - it s ;. Write the values f(p) In polar coordinates,
f(p) = r(P)-exp{i-,9(P)}, -n < 9 s it.
If the two series s1(f) = P p-1
(f(p) - I), S2(f) = P P-1
If(P) - 112
are convergent, the following five series converge: 1 = Z p-, -'5(p), P
The Theorems of Elliott and Dabouss1
244
Ell = P p 1.82(P), p-1
III = P
IIV =
IV =
21
1
P
P
log r(p), log r(p) 12,
for any q 2
P-1. (l r4(p)
P
1.
Proof. Clearly, 3 s r(p) 5 q, and cos(9(p)) 2 z /3, and so - 6 n < D(p) < 6 7t, Taking real and imaginary parts, the convergence of the two series S1(f) E4, where and S2(f) implies the convergence of the four series E11
E1=IP1' P P-1
2
9(p)
{ r(p) sin 9(p)
.
},
P 3 =
P
r(p) -COs 8(P) - 1 }2 =
P-1
= P P-1. {
2'(r(p)'cos4(p) - 1) - 1},
and
P -I ==
P
8(P)}2.
'
The inequality r2(p) 2 9/16 implies ES =
P -I
.
{ 1 - cos2(8(p)) } < ao.
P
Throughout the interval - 6 < 9(p) < 6 the relation
1-cos24 '
7C
02
1 - cos2(8(p)) 2 N
2
0.5.8
-0.F
-0,5
0.4
0
-0 a
.i
0,
C1
7
01
0S m
holds
with
a
suitable
positive constant y. This implies the convergence of the series
Figure VII.1
III =
p-1
P
.
32(p). 2
The relation show that (3.1)
cos
and the convergence of
'D
1
P
ao.
11
= P
PP)
ylI.3. Multiplicative Functions in B
245
is Z p-'-( r(p) - 1 ), and so it
The sum of this series and of
P
is
Similarly, starting with (3.1) and utilizing the convergence of G3 and 2: 4' we find that convergent.
IP p-'
r(P) - 1 )2 < co.
Since
r9 - I = q (r-1) + 0((r-1)2) In < s r s
+
1
,
the series 2: V is convergent; the approximations
log r _ (r-1) + O((r-1)2), log2r
=
0((r-1)2), a S r s
1
+ e,
imply the convergence of ZIII and ZIV' Finally, sin $ = $ + Q;32) 8(p). Together with the CAUCHY-
gives the convergence of E p-1 r(p) P
SCHWARZ estimate (21p1
'
I r(p) - 1 I
P
Is(P)I )2 s E P ' ' (r(p)
P-1. 42(P) < 00
t )2 P
P
we obtain that
I = EP p-'
'
p-'
r(P)'
'
P
(r(P) - 1)',8k(P)
is convergent, and the lemma is established.
11
Proposition 3.2. Assume that f is a strongly multiplicative arithmetical function, for which the two series
S1(f) = E P-1f(P)-1) P
and
S(f) 2 = PI
p-1'If(p)-112
are convergent. Assume, furthermore, that for all primes p the condition I f(p) - 1
is satisfied. Then f e $1 and
II f II
I
9
Se
< oo
for any q 2 1.
Proof. 1) First we obtain II f Ilq < CO' using RANKIN'S trick (11.3). Recall that f is strongly multiplicative and satisfies If(p)I s 4 Therefore,
The Theorems of Elliott and Daboussi
246
log x
s2
< n s x If(n)Iq
5 2- Z
nsx
log pk = 2 I log pk
If(n)Iq
I msx
log n
f(n)
<nsx
f(m)Iq
.
I
log pk K,
in the notation of Lemma 3.1. Making use of the inequality eZ -
1
=
I
I f z e' do
1
I
0
s
IzI
max { 1, eRe z } S IzI .(1+ IezI)
we obtain
TI ,K f(p) - I
Pi
i
=
I ew(n) - 1 s
I w(n) I . (1 +
Starting with AN s N-1
If*(n)I
EnsN Iw(n)I CAUCHY-SCHWARZ inequality gives A
N
s
' E Iw(n)I2 { N n5N
} i. 2 { (
t
N nsN
If*(n)I2 1+ )
(
ew(n) I ).
the
+
1'
w(n) 2
N N If (n)'eI
}
A(')}',2{(A 2)) +(A 3))}. First it will be proved that lim supN--> m AN(2) is bounded uniformly in
K. Using the 9p-evenness of fwe obtain A (2) = N-1 N
.
nsN =
N-1
If(gcd(n,P))12 = N-1
If(d)12. {w(a )'
d where IOI s 1. The error term is N dT a
d
+ ®'
If(d)I2
rnsN/d, m,T/d)=1
1
21, a
If (d)12 = N P TI {1 + 1f(p)V } 5 N . 26)(5p)
The main term is QN21) = P-1. { 1f12 * cp }(9)) = IT P-1 p &K K
{ cp(P) + If(P)I2) = IT { I + K
If(PP)I'-1
I.
The Theorems of Elliott and Daboussl
248
[The star * denotes convolution, see Chapter I, Section 1]. The inequality I + x s e", valid in -oo < x < oo, and the convergence of Ev_
I p-1 { If(p)12 (from Lemma 3.1) imply P
A (21) s exp (
where
the
lim supN---)
.
ON )
{ If(P)I2 - 1 }) s C3 < oo,
P-1
C3 can be
bound (2
I
psK
chosen
independently
of
K.
Thus
Is bounded.
f(n), and using the CAUCHY-SCHWARZ Observing that inequality, one immediately obtains llm sup A (3) = II f II2 < w N ->m
(by part 1). The proof of Proposition 3.2 will be concluded by showing AN) 0 as K -co. oFirst lim supN AN) S N n N I
P
W(n)
Pp)
s 2 C1
+2
w(p) I2 =
I
P
PSN
The TuR.AN-KUBILIUS inequality N > K, ON11)
I2
K m
Proposition 3.3. Let q 2 1, and let and
II f II
q
f
E
6'
9
11
be multiplicative. Then f E 2
< oo.
Remark. From VI. Theorem 2.8, the finiteness of II f II and the fact q that f c 21 imply f E $r for every r In 1 s r < q. In fact, for multiplicative functions f with mean-value M(f) $ 0, the stronger conclusion f 2q is true. This will be shown in this chapter, Theorem S.2. E
Corollary 3.4 [H. DELANGE]. If f is multiplicative, if the series S1(f) = E P-1
'
{ f(p) - 1
}
pll.3. Multiplicative Functions in S'
249
converges, and If 1fl s 1, then the mean-value M(f) exists and f is $1. In fact, 1 1f1 1 q < oo for every q a 1, and so f e ,X19 for any q Z 1. in
This follows immediately from Proposition 3.3. The estimate p-
f(P) - 1
If(P) - 112 s p-1 If(p) - 112 + P-1 (I - If(P)12
_
fp-1 P
P
and the convergence of S1(f) imply the boundedness, hence convergence of S2(f). Therefore, f e 92, hence f ,$1, The finiteness of II f 11q is obvious from the estimate IfI s 1. E
The Proof of Proposition 3.3 is achieved by an application of the Relation-
ship Theorem of Chapter III, which enables us to reduce the assertions of Proposition 3.3 to Proposition 3.2. 1) Let f satisfy the assumptions of Proposition 3.3. The convergence of the series Z p-1 ( f(p) - ) implies the existence of a constant L 2 3 with the property 1
I
- (P -1 ), if p z L.
f ( p ) I< 2'
Define a strongly multiplicative function f by f(p), if
f(p) - 1
I
I
s A and p i L,
otherwise.
1
The functions f and f* are related. j = 1, 2 one has 1
E P - If(p) - f*(P)IJ 5 If(p -1I>i
s4 .Z
P-1
p-1
p
.
fact more is the case: for
In
.
If(P)
If(p)
IJ
+ psL E
P-1
- 112 + E P-1 psL
.
-
If(P) - 1
If(p) -
1
lJ
IJ
s-y(f,L) K, are convergent; therefore the summands tend to zero if K -3 co. Thus lim 11f - f"11 q 0, and the theorem is proved. Recall
K-4 oo
q
4.B. A second proof for Theorem 4.1. In the special case
q = 2, f z 0, f completely multiplicative, M(f) $ 0, a simple proof for Theorem 4.1 is available. We calculate the RAMANUJAN coefficients of f and use BESSEL's inequality.
Theorem 4.3. If f
01 is completely multiplicative and has a mean-
e
value M(f), then: 1) For every r
e EN
ar(f) _
(f*µ)(r)
M(f).
2) If in addition M(f) $ 0, then the map r H {M(f)}-1 ar(f) is multiplicative and apk(f) = (P-11'f f(P) - 1} .
-{M(f
(f(P))k1
VII.4 Multiplicative Functions in .Sq 3)
If M(f) $ 0 and a,(f)l2
r=1
11f112 < co, then
p(r) = II P p-plf(p)12
s IIf IIZ.
proof. 1) Using the representation of the RAMANUJAN sum cr as a sum over the common divisors of n and r, we easily obtain
(f*µ)(r)
M(f).
2) The function r N ar(f)/M(f) = (f *V)(r)/cp(r) is obviously multiplicative, and the values of the RAMANUJAN coefficients at prime-powers, given in the assertion of the theorem, are easily checked. 3) BESSEL's inequality implies that A(f) = IM(f)l2
I(f µ)(r)12
r=1
=
r=1
Iar(f)12 w(r) s IIf11Z
is finite. From Lemma 1.6 we know that M(f) = IT (p - 1) Therefore,
A(f)
_
{ p - f(p) }-1.
P
J p
IP-112
1
Ip-f(p)12
P
+
P-1
If(p)-112 p-If(p)12
IT
p-1 P-If(p)12
We are now going to prove the following special case of Theorem 4.1.
Theorem 4.1'. If f
91 Is completely multiplicative and non-negative with mean-value M(f) $ 0, and if 11 f 112 < oo, then f e $2. e
Proof. First we show f2 c g the series S1(f2) converges, since f2(p) (f(p)- 1)2, and S1(f) and S2(f) do converge, and + = 2' (f(p) - 1) 1
SZ(f2) = O(SZ(f)) = 0(1). The convergence of the other two series follows from BESSEL's inequality. Y(Pk) < 11f112
.'t 1 kZ
IM(f)1-2 < oo.
P to Theorem 4.3, every summand has the form According f( )-12 f
p- 1
k-1
2
Pk-,
Using x2 s S. (x - 1)2, if Ixl z 2 rS- , we obtain
I
If(p)j>
Y. If(p )I k>1
Pk
< 00.
The Theorems of Elliott and Daboussl
254
This series is a majorant of Sz 1(f2) and of S3,1( f2 ), so these series are also convergent. From f2 e 91 we conclude f2 a ,$1, by Theorem 3.3; and f from VI, Theorem 2.9.
E
.`82 follows 11
4.C. Proof of Theorem 4.2. The multiplicative function f is factorized as f = If I h. We prove a criterion, ensuring that IfI is in &q and h E $q (Theorems 4.5 and 4.7). In
the case where M(f) * 0 these results give a criterion for the function f to belong to Sq. Lemma 4.4. Let f E A1, and assume that M(Iflr) exists for some r 2 0.
If M(f)*0, then M(Iflr)*0. Proof. First Ifl E A1, M(Ifl) exists and y = M(Ifl) z IM(f)I > 0. Choose a trigonometric polynomial t E A near Ifl, so that Ifl - t ll, < I ' Y. Then, for x Z x1, 1
II
S = X_'...' 2
12
If(n)l = x-1. E If(n)I - x-1' Fr If(n)I nsx nsx, (n)s1Y
_
1
nsx
Y
IltllUl,
hence
If(n) Ir z (, Y)r
This proves Lemma 4.4.
Y'
0. 11
Theorem 4.5. If f e ,81 is multiplicative with non-zero mean-value M(f), and if II f Il < oo for some q 2 1, then Ifl E ,$q. 9
255
VII.4. Multiplicative Functions in $q
proof. The function g = 2- norm is 1 s r < q. Hence:
IIgh12 =
Ifl, is non-negative and multiplicative; its Jar, if (II f lhq)3q < co. VI, Theorem 2.8, gives Ifs
,$'q and g e $1 (see VI, Theorem 2.9); if 1 5 q < 2, then IfI E $1, and so g e ,$2/q c $1, by VI, Theorem 2.9.
if q 2 2, then IfI
e
In accordance with Lemma 4.4 the mean-value M(g) is non-zero. Therefore, Proposition 2.1 gives g E g2. This easily implies g2 a 91, since S1(g2) = 2
-
S1(g) + S2(g), S2(g2) = d( S2(g)),
SZ 1(g2) = d(S2(g)), S3,1
using the notation of Definition
1.2.
(g2)
= S3,2 (g),
By Proposition 3.3 the function
jfjq = g2 is In .Y31, hence Jfl E $q.
Now, for an arithmetical function f, define a function h by f(n)
If(n)I
if f(n) $ 0,
0
if f(n)=0.
h(n) = ,
If f is real-valued, then h is the sign-function. A first result on this "generalized sign-function" h is the following proposition.
Proposition 4.6. Let f be an element of Y31. If there exists a constant S > 0, for which the upper density dens { n; lf(n)I < 8 } = Ilm sup x-1
X ->-
then the function h is in
$1
.
tt{ n 5 x, lf(n)I < 8 } = 0,
again.
Proof. The function s8 : C -) C will be defined by if Izl a S,
s8(z) = if Izl < S. Then
The Theorems of Elliott and Daboussi
256
and so s8 is LIPSCHITZ-continuous; by Theorem VI. 2.11 (2) the composition sso f is in .1, and h is II. 111-near sso f on behalf of Y_
nsx
I
h(n) - s8(f(n))
X
=
I
nsx,If(n)I 0, and Proposition 2.1 yields IfI' E 02.
proof of Theorem 4.2. Write f = h Ifl. First of all, by Proposition 3.3 and Theorem 4.5, the function IfI is in 21q. Next, the function h is in 21 according to Theorem 4.7, The function h is bounded; therefore VI, Theorem 2.6 (iii) gives h
E Sq.
IfI
VII.S. MULTIPLICATIVE FUNCTIONS IN Aq
WITH MEAN-VALUE M(f) * 0
The aim of this section is to prove the following theorem. Theorem S.I. Let f be a multiplicative arithmetical function with meanvalue M(f) * 0. Assume that q z 1. Then the following four statements are equivalent: (1)
fEeq,
(2) (3) (4)
f E Sq,
fDq, f
Aq.
If q 2 2, then (1) to (4) are also equivalent to
(5)
II f 11
q
< Co.
The Theorems of Elliott and Daboussl
258
Furthermore, In any case and for every prime p, 9F
(p, 1)
= I +
P-1.
P-2.
f(P) +
f(p2) + ... * 0.
Remark. Without the assumption M(f) $ 0, the implication (4) wrong, as may be seen from the example following Lemma 1.4.
(1)
Is
(2) is contained in Theorem 4.1. The ImpliProof. The implication (1) cations (2) (3) (4) are trivial. The implication (4) (1) will be proved in two steps: first the convergence of S2 q(f) and S3 q(f) will be shown,
then that of S1(f) and of SZ(f) by relationship arguments.
1) We consider the function g = Ifl"q and use Proposition 2.1. Since '4q, Corollary VI.2.10 gives g E 42. The mean-value M(g) is nonIfi zero for M(Ifl) z IM(f)I > 0 and Lemma 4.4. Proposition 2.1 shows that the three series E
St(g) _ Zp P 1 S2(g) _ Zp p-1
(g(p) - 1),
-
.
I g(P) - 112,
S3,2 (g) = Ep I kit p-k lg(pk)12 are convergent. So S3 (f) = S3,2 (g) is convergent. q
From the convergence of S2(I f11q) we obtain (S.1)
p-
Zp,lf(p)I>5/4
.4 m
and
2: p
If(p)I 0 (see
Exercise 5).
VII.6. MULTIPLICATIVE FUNCTIONS IN A q WITH NON-VOID SPECTRUM
If
f is an
arithmetical function
f(p) = M(f) exist for every
(i
its FOURIER-coefficients !R, and its FOURIER-BOHR specin A 1,
E
trum is defined as spec(f)
E
iR/7L:
lim sup
I
x -) m
x1
I
Ix
> 0}.
Remarks. 1) For functions in Al the Jim sup in the definition of spec(f) can be replaced by limx _ - this limit exists. 2) For every arithmetical function f the condition M(f) $ 0 implies spec(f) * 0, and this implies If II1 > 0. H. DABOUSSI proved the following theorem in 1980.
Theorem 6.1. Let f be a multiplicative arithmetical function, and assume
that q 2
1.
(I) If
f
(D.1)
E
.94q,
and spec(f)
(D.2)
then there exists a DIRICHLET character X, such
(D.3)
S1(X.f),
are convergent.
S2'(X'f),
SZq (f),
that the four series
and S3,q(f)
The Theorems of Elliott and Dabogssi
262
(II) Conversely, If the series (D.3) are convergent for some DIRt CHLET character X, then f e Dq.
Corollary 6.2. If f is a multiplicative function with spec(f) * A and if q Z 1, then the following three statements are equivalent. (1) There exists a DIRICHLET character X such that the four series given in (133) are convergent.
(2) f ,v4 q. (3) f E Dq. E
First we give a variation of DABOUSSi's result 11.6.2.
Lemma 6.3. If f E Al is multiplicative, then, for every irrational (i
the
FOURIER coefficient ?(a) = M (f e_13 ) is zero.
Proof. Without loss of generality, assume that M(Ifi) > 0; otherwise A
I f (p)I = 0 because of I f (R)I s M(IfI). Theorem 5.1 shows that
IfI
is in
Of Choose a prime po so large that for all primes p > po
If(P)I s 4), and k = if p > po, If(p)I s y , and k = 1, if k 2 2.
1,
The functions f and F are related:
Z p-1 P
'
If(P)-F(P)I = O(S21(If1)) = C7(1).
Both of the functions f, F are in
Ip-1.f(P)I2 ZP i 1p k22
=
since
0(1) + 0(S21(Ifl)) = O(1), 53,1(IfI) = 0(1).
The estimate 9 ( p, 1) - 11 = p-'- IF(p)I s 4 p- < 1 shows that F is . The Relationship Theorem 111.7.1 (or III, Exercise 9) gives, with a in multiplicative function h, satisfying f = h * F and Z n-1 Ih(n)I < ., 1
,,II.6 Multiplicative Functions in Aq with Non-Void Spectrum
n=1 n 1 h(n)
f (a) _
263
a)
F(n
for every real a. The values IF(p)I are bounded, therefore DABOUSSI'S Theorem 11.6.2 gives F(a) = 0 for every irrational a, and thus equation (6.1) gives the statement of Lemma 6.1 as soon as IIFII2 < CO is proved. By II, Theorem 3.1, (3.3)
x-'-Y- IF(n)12 = of exp{ nsx psx,lf(p)ls5/4
P
1'( If(P)12-1) })'
The last series is seen to be convergent, using x2 - 1 = (x- 1)2 + 2 (x - 1 ), and
Z
If(P)Iss/4
P-1' ( If(p)I - 1 )2 = SZ (Ifl) = 0(1), P-1 .
If(p)
s5/4 P-1
jf(p)>s/4
(If(P)l -1) = s 1( IfI) -
.(
If(p) >5/4 p-1
If(P)I - 1)
If(p)Y_
,
5/4
P-1. (If(P)I
-1 ),
If(P)I = s'1( Ifl) = 0(1). D
Next we prove Theorem 6.1 (I) in the special case where f is completely multiplicative.
Lemma 6.4. Suppose f is completely multiplicative and q z 1. If f E.P41
(D.1)
and spec(f)
(D.2)
then there is a DIRICHLET character X for which M(X'f) * 0.
Proof. Assume that M(X'f) = 0 for all DIRICHLET characters X. The calculation of the FoURIER coefficient f (p), where R = ra r z 1, is rational, is achieved in the following way: x-1
Y nsx
x-1
f(n)'e-a/r (n) =1 psr e-a/r(P)
dir
lspsr
'
(P) e -a/r())
gcd(p,r)=d _
21 nsx, nap mod r 1
X
f(n))
21
nsx
nap mod r
( fd)' /r' P x/d d 1spr' `e -a()
gcd(p',r')=1
f(n)
x/d f(
map' mod r'
The Theorems of Elliott and Dabou881
264
with the abbreviations p' = p/d, r' = r/d. The orthogonality relation, for the characters X mod r' [ in case gcd(r', p') = 1 ] imply
f(m) _ (r')
X
m:r X
m-p' mod r'
1x msx X(m)'f(m) = 0(1),
X(P')
X mod r'
0 for every character X. Thus we obtain by our assumption f ((i) = 0 for every rational number p. Since spec(f) C Q (see Lemma 6.3), we have a contradiction to (D.2), and Lemma 6.4 is proved. 11 Proof of Theorem 6.1. I) Our goal is to show that, given a multiplicative function f in .049, where q > 1, with non-void spectrum spec(f), there exists a DIRICHLBT 0. character X such that Given f e 4q with non-void spectrum, then Ifl a .049 and M(Ifl) > 0. Therefore, we deduce Ifl a 9q , using Theorem 5.1. In particular, the series
S' q(Ifl) = S'
(f),
and S3 q(Ifl) = S3,q(f )
are convergent. The convergence of these series enables us to choose an integer P with the properties 0
IP ''f(P)I < < , and Xk21
P-k
If(Pk)I < z for any p z Po.
Define the "nearly-completely" multiplicative function f* through f(pk), if p S Po1 f*(Pk) =
{f(p)}k, if p > Pa. *
Then f and f are in ;, and cp f(p,s) * 0 in Re s z for every prime * p > P0. Furthermore, f and f are related, and Theorem III, 7.1 admits 1
the conclusion f* = f * h, where I n=1
therefore f * is in .J1
n-1
Ih(n)I
1. Then prove: the existence of the mean-value M(f) $ 0 implies
the existence of M(d)(f) = limx
x-1
-> oo
1nsx,gcd(n,d)=1 f(n)
if d is composed solely from sufficiently large primes, and M(d)(f) =
{Wf(P,1)}-t
pid
2) Define an arithmetical function Xr,d in the following way: Xr,d(n)
11, if gcd(n,r) = d, jl
0, if gcd(n,r) $ d.
Prove for any function f E $1: (a)
11f1l1
> 0 if and only if there exist positive integers r, d such
that M(fXr,d)$ 0.
(b) If f is multiplicative in addition, then
11f1I1
there exists a DIRICHLET character X = Xr 1
> 0 if and only if for which M(fX)* 0.
267
VII.7. Exercises
3 ) If f is multiplicative, q z I and
I I f1 1
f E'q 4 f
q E
> 0, then prove that S q.
4) If f E $1 is multiplicative and IIfII1 > 0, then the function h(n) =
f (n)
If(n)I'
if f(n) $ 0, h(n) = 0, if f(n) = 0,
is in A31.
5) Let f be multiplicative, and assume that q z 1, and II f
11q
II f III >
0. If f is in
.Y31,
< oo, then f e .$q.
6) Given d E IN, define the function i by p(n) = if gcd(n, d) = 1, and p(n) = 0 otherwise. Assume that f is multiplicative, q z 1, f 11q < 00, 1
1171
and - for every prime p - the sum zk> O p-k prove that Iif IIq < Eb.
I f(pk) I q < oo
.
Then
269
Chapter VIII Ramanujan Expansions Abstract. In this chapter, for given classes of arithmetical functions, mean-values and RAMANUJAN coefficients a(f) = {p(r)}'. M(f cr) are calculated, and the convergence properties of RAMANUJAN expansions are studied. To achieve this, It Is advisable to deal with mean-values Md(f) = lima - m T-nsx,n=0 mod d f(n) of arithmetical functions In residue-classes. Rather simple criteria use the ERATOSTHENES-MdBIUS
transform f = f * µ. Better results are obtained when the results of Chapter VII are used to obtain information on mean-values Md(f) and RAMANUJAN coefficients ar(f). For multiplicative functions in A2 the RAMANUJAN expansion I a r(f) Cr (n) is pointwise convergent. Finally, still another proof of PARSEVAL's equation Is given for multiplicative functions In A2.
Ramanujan ExpansioA$
270
VIII.1. INTRODUCTION
The RAMANL[IAN sums cr, r =
..., were defined in chapter we shall utilize both of the representations 1,
2,
cr(n) = Z dlgcd(r,n) d - µ(r/d) =
I,
§3. In this
lsasr,gcd(a,r)=1 exp(2,1 r n),
and the multiplicativity of the map r H cr(n). Due to the orthogonality relations for RAMANUJAN sums,
cp(r), if r = s, and
0 otherwise
(see I, Theorem 3.1), for an arithmetical function f we expect a RAMANL[IAN expansion
f ti -7 r ar
(1.1)
c
r
where the coefficients ar = ar(f), in the case of the existence of the limits involved, are given by (1.2)
ar(f)
(p(r))-1
.
(qq(r))-1
=
'
,
using the inner product notation = There are many examples of arithmetical functions possessing a [convergent or not convergent] RAMANUJAN expansion (1.1): the coefficients (1.2) do exist, for example, for all functions In A 1. There are different concepts of "convergence" of the RAMANL[IAN expansion. In VI.4 for functions f E SZ the relation lim
R--> c,
11 f - E
rl RI
ar(f)' Cr 112 = 0
was proved (a still better result is provided by PARSEVAL's equation), and in VI.S we proved, analogously, that lim
R-) o
11f- E
rI Rl
ar(f)'crII = 0
for functions f in S1. A rather trivial example of the convergence of series with RAMANL[IAN sums, but with "wrong" coefficients, was given
In Chapter V, Theorem I.I. The difficult question of polntwlse convergence of expansion (1.1) for a "large" class of arithmetical functions
VIII.2 Wintner s Criterion
271
was dealt with in Chapter V: Following A. HILDEBRAND, it was shown (V, Theorem 1.1) that the RAMANUJAN expansion of any function f in
is pointwise convergent. Many special examples of functions with pointwise convergent RAMANI.uAN expansions are given in HARDY's paper [1921].
R. BELLMAN [1950] suggested the deduction of asymptotic results for such sums as 2:nsx f(P(n)), where P is an integer-valued polynomial, by using the [convergent] RAMANLUAN expansion of the arithmetical function f to be investigated. However, in order to obtain good results, one has to have intimate knowledge of the convergence properties of the RAMANUJAN expansion, and so this approach may not be very promising. Unfortunately, this method does not work for f = µ2, for example.
V I I I.2. WINTNER'S CRITERION
A first general and simple result is due to A. WINTNER. It has the advantage of being valid for every arithmetical function, satisfying condition (2.1), which unfortunately is rather restrictive. On the other hand, the assumption of multiplicativity is not needed.
For any arithmetical function f the function f' = µ*f is called its ERATOSTHENES-MOBIUS transform.
Theorem 2.1. Assume that the ERATOSTHENES transform f' = µ * f of an arithmetical function f satisfies the WINTNER condition (2.1)
E
n-1 n=1
'
If'(n)I < CD,
then:
(i) The function f is in S (ii) Its RAMANUJAN coefficients (1.2) exist and are equal to
ar(f) _
1sd<m,d-O mod r
d-1
f'(d).
Ramanujan Expansions
272
(iii) The FoLrRIER coefficients
f (r) = M( f e_a/r ) exist, and f (r) = ar(f) If gcd(a,r) = 1. So the FoURIER coefficients do not depend on a. (iv) If, moreover, the series (2.3)
n=1
2w(n)
.
n-1
.
If'(n)I
Is convergent, then the RAMANUJAN expansion : Oro= 1 ar '
Cr(n) = f(n)
is pointwise convergent, and its sum has the "correct" value f(n): X- 1 ar ' cr(n) = f(n) for every n E N.
Remark. In fact, the method of proof of (iii) gives a stronger result: if M(If * iLl) = 0, and f e 941, then f (r) exists and equals
f (r) = In=O mod r n-'' (f *µ)(n), if gcd(a,r) = 1. The condition M(If * µI) = 0 is satisfied, for example, if f is in 0q for some q > I (Theorem 3.5). Proof of Theorem 2.1. (1) and (ii). The function fK(n) = Ed!n dsK f'(d)
even mod K!, and so is in 21. We expect that fK is "near" f: n H GdIn f'(d), if K is large. Using (2.1), the norm estimate is
Ilf - fKll l
s lim sup x-1 z E I NO s Z If'(d)I 1 --* 0, as K -> co, nsx dln,d>K d x -4 m d>K
shows that f E $1. Therefore, the RAMANUJAN coefficients ar(f) exist. Next, cp(r)
ar(fK ) = _
lim
x -4 m dsK
x-1
f'(d)
'
z
I
lim
x
nsx d!n,dsK
x -> m
1
'
f'(d)
c (n) r
c (n)
nsx,n,o mod d r
Y (r ) F, mod r f ' (d) d dsK,d-O as is easily shown using the representation of cr (n) as an exponential sum (see Exercise 2). The estimate 1.
VIII.?. Wintner's Criterion
273
lar(f) - ar(fK)I 5 (cp(r))-1
sup I cr(n) I s II f - f Kill
II f - fKlll
yields, by letting K -4 oo, the truth of (2.2). (Iii) Assume that gcd(a,r) = 1. Put f = depending only on r, x
1 E f(n) e- /r(n) nsx
x-'
=
nsx
e_
I
a/r (n)
* V. Then, with O-constants
dn
f'(d)
1 z e(- ra'm) msx/d
=dsx E
z
I If '(d)l . O(1). f'(d) x (Xd + O(1)) + x1dsx 1
dsx,d-0 mod r
The absolute convergence ofax d-1 f'(d) gives a x1f'(d)I = o(x) [by partial summation], and the formula for the FoURIER coefficients follows. in A1, then the FouRIER coefficients exist. Therefore, If M(If'l) = 0 is assumed, the last displayed equation gives that
If f is
lim
x -9 m dsx,d=O(r)
f'(d)
d-1
exists and equals f (a/r).
(iv) Using (2.2), and
c (n) = d, if din, c (n) = 0 otherwise rd r rd r (see Exercise 3), we estimate the difference D
R
= f(n) -
21
a (f)
cr(n) =
rsR r
d
c (n) r
f'(d)
d-1
(d - rjd21
T c (n)) = Z rld,rsR r d
d-1
cr(n)
f'(d)
Thus we obtain IARI s
The map d H Fr
rd
rTd
I
Cr
(n) l
d-1
.
If'(d)I
Icr(n)I = IT E
pRlld Osks
This gives
Icr(n)I.
Icpk(n)
s IT(1+cp(p)+ ... pld
'
rla is multiplicative, and so, if pm ll n, dZR
+
cp(pm)+ pml
c (n).
rld,r>R r
2n'(d).n.
274
Ramanujan Expansions
AR) s n- dZR d-1
If'(d)I
24'(d)
_o,
as R oo, and the convergence of the RAMANI.JJAN expansion value f(n) is proved.
to the 13
Examples. We mention the [absolutely convergent] RAMANUJAN expansions n-1
n-1
where 92 (r) = r2
a(n) =
p(n) = 6 IC-2
1
r2
n2 .
z°r°
µ(r)
cr(n), {T2(r)}-1
.
cr(n),
IT (1- p-2) (Exercise 4). Several approximations of pjr
cp(n), by partial sums of its RAMANUJAN expansion, are given in Figure 1-6. Abbreviate 6 7-2 E r5R µ(r) (p2(r))-1 cr(n) by SR(n).
n
Figure VIII.1
Values
of
SS(n) in the
range
I s n s 600. 100
200
300
600
Figure VIII.2 Values
of
S1O(n) in the
range I
s n s 600.
Figure VIII.3 Values of S2O(n) in the range
1 s n s 600. 100
200
300
600
VIII.2. Wintner's Criterion
275
Figure VIII.4 Values of S4O(n) in the range
I s n s 600.
Figure VIII.S Values n
-1
.
cp(n)
of In
the range
I s n s 600. 100
200
300
600
Figure VIII.6
Values of SI (I= S, 10, 20,
40), and of n Hn-1 cp(n).
Figure VIII.6 gives the values of S5(n), S1O(n), S20(n), S40(n), and -1
n
cp(n) [in this order] in the range I s n s 120.
Hopefully, these diagrams give an impression of the convergence of partial sums of the RAMANLUAN expansion to the function n -1 p(n).
Ramanujan Expansions
276
VIII.3. MEAN-VALUE FORMULAE FOR MULTIPLICATIVE FUNCTIONS
RAMANL[IAN coefficients are closely connected with mean-values on residue-classes; In order to calculate these coefficients, we need some mean-value formulae, particularly for multiplicative functions. For the sake of completeness, we repeat some results from Chapter VII. Lemma 3.1. Assume that f is a multiplicative arithmetical function with finite semi-norm 11 f 11 for some q > 1. Then If(n)I s C n1,'9 for some constant C > 0, and f(n) = o(n1/q ), as n - oo. P-2 . If(p)I2 < oo.
(a)
(b)
P
2:
(c)
2:
p km2
X Z p k22
(d)
P-k' If(Pk)I < oo.
p-k If(pk)Ir < oo for every r In 1 s r < q.
For the proof see VII, Lemma 1.4.
Proposition 3.2. Let f be a multiplicative arithmetical function, with a mean-value M(f) * 0. Assume, further, that the series
Z P-k.If(pk)I
(3.1)
k=0
Is convergent for every prime p. Then M(f) = urn a
x-1(0) 1+
E on =
n=1
=
o
llm>
a-lim1+ np (1 + (I
1+
+
pto
p
f(p)-1 pa
P
+
f(p2)-f( p)
+
P
Therefore, for every prime p, 1 + f(p)-1
+
p2
p
Proof. The first assertion
f(p2)-f(p)
is
+
...
$ 0.
the continuity theorem for DIRICHLET
,VIII.3. Mean-Value Formulae for Multiplicative Functions
277
series, which is a simple application of the formula for partial sumis convergent for mation. In particular, the DIRICHLET series n=1 n no > 1. The second assertion is obvious from the first and the assumption M(f) * 0.
Remark. Assumption (3.1), for the convergence of E p-k If(pk)I, is fulfilled, for example, If f E Aq, where q > 1, and M(f) * 0, or if f Is in (see Chapter III), or If En I(µ*f)(n)I < co, or if f E A1, and M(f) 0 (VII, Thm. 5.1). Theorem 3.3 (Formulae for mean-values). Assume that f is a multiplicative function In 9q, where q > 1, with mean-value M(f) * 0. Then
II( 1+ P
M(f)
f(p)-1
f( p2)2f
+
p
+... ).
p
In particular: (a) If is completely multiplicative, then M(f) = III 1 + ( f(p)1 ) (1 -
f (p) )-1 1.
(b) If Is strongly multiplicative, then
M(f) = II
l
I +
RP)-1
).
(c) If is 2-multiplicative (this means that f(pk) = 0 for every prime p and every exponent k z 2), then
M(f) = IT ( 1 -
1+
f(p)
).
Proof. We use the formula of Proposition 3.2. Well-known results on infinite products (see the Appendix, Theorem A.7.1) guarantee that the main assertion of Theorem 3.3 is true as soon as the convergence of the three series 12 Z P-1' ( f(p) - 1 ) , Z P-2' 1 f(p) - 1 ,
P
P
and PEkk2 E P-k '
is proved. But this is obvious from the assumption f E
I f(Pk) E; q.
f(pk-I)
11
Remark. The assumptions f .vlq, q z 1, f multiplicative, and M(f) * 0 imply that f e (see Chapter VII, Theorem 5.1). E
q
Ramanujan Expansions
278
A simpler result is the following theorem.
Theorem 3.4. Assume that f Is strongly multiplicative, M(f) * 0 exists, and II f II < co for some q > 1. Then f(p)-1
+
M(f) = IT P
Proof. We start with M(f) =
( 1 + p-° ( f(p) -1) ). The pro-, 1+ 1T P
lim C3
duct is equal to
P
ll p-2a. If(p)-112 )J)'
exp (E {p-, . (f(P)-1) + P
The series Z
p-2. If(p) - 112 is convergent. Therefore, P-Q
limo .. 1+
'
( f(p) - 1 )
exists. In the same manner as used in the proof of the DABOUSSIDELANGE Theorem in Chapter VII, this implies the convergence
of
Z p1 (f(p) -1), and the desired result follows (see the Appendix). P
11
The following theorem deals with the ERATOSTHENES-MOBIUS trans-
form f' = µ * f of an arithmetical function f. Theorem 3.5. Assume that f e 9q Is multiplicative, and q > 1. Then: (a)
The mean-values of f' and of If'I are zero.
(b)
The series Y ,'=i n-1 f'(n)
converges, with limit M(f).
Proof. (a) It suffices to show that M(If'I) = 0. We use Theorem 3.1 from
Chapter
Ey I f'(pk)1
Ps
E 1 f'(Pk)I
P`sY
II
to
estimate
21
The
I f'(n)1.
nsx
assumption
log pk s c1 y is satisfied: log pk
pksy
f(pk) -
f(pk-1)1
.
log
plc
Z loge+3 Ep°syIf(Pk)logpk
p sy
5 0(y) + 3( E If(Pk)Iq)1/q .( E (log
pk)q')1/q'
P°sY
= O ( y + y (log Theorem II, 3.1 (3.2) gives
Y)1-1/q')
=
O( y ' (log Y )1/q )
J
VIII.3. Mean-Value Formulae for Multiplicative Functions
279
x Ensx If'(n)I x)-1+1/q
s c2 (log
exp {psx E P 1, If'(P)I + psx E k22 E P-k If'(Pk)I }
The sum E
E P-k If'(Pk) I
psx ka2
52E
S
EP
psx k22
is pEx
bounded P-1
If(pk-1)I )
E P-k (If(pk) I+
E
psx kk2
k, If(pk)I + E P-2. If(P)I = d (S3,9 psx
+
S2,q + 1)
in x, and
If'(P)I S {
P-t , I f(p) _ 1I2}2
Psx,iffp)kSS/4
{ Px
P-' } 2
+ n(SZ,q)
= O ( log log x ).
Therefore M(If'I) = 0. (b)
x-1,E nsx
f(n) =
nsx d n f '(d)
E dsx d
f'(d) = x-'-dsx E f'(d)'(d + n(1) )
+
(7(x-1
d xI
f'(d)I).
The existence of M(f) and M(If'I) = 0 now imply the convergence of E d_'- f'(d) with limit M(f). Remark. If f Is strongly multiplicative, we can also prove the convergence of the series n-'- f'(n), for r = 1, 2, ... n-0 mod r Define multiplicative functions by k X (P ) =
r
1if 101,
if plr,r,
F(
pk)
=
f(p ), {
1,
if p4 r , if plr.
Then the convolution relation F = 1 * (f' Xr) gives
x 1 E F(n) =dsx E
d-1
f'(d)Xr(d) + d(
x-1,
If'(d)Xr(d)I ).
E dsx F Is In 6q, the mean-value M(F) exists, and M(If' XrI) S M(If'I) = 0; so we obtain the convergence of Egcd(d,r)=1 d-1 f'(d). But
nsx
Ramanujan Expansions
280
f (n) _ (r 1 f'(r))
2:nm0 mod r n
d-1 f (d),
2: gcd(d,r)=1
because f Is strongly multiplicative (and f' Is 2-multiplicative). Thus the assertion is proved. o
VIII.4. FORMULAE FOR RAMANi1JJAN COEFFICIENTS
There
is
a
close
ar(f) _
connection
between
coefficients
RAMANLIJAN
and mean-values on residue-classes, Md(f) =
lim
X ->
ao
x1
nsx,n®O mod d
f(n).
Proposition 4.1. For every arithmetical function f,
(1) the existence of all the mean-values Md(f) for d =
1,
2, ...
implies the existence of all RAMANLUAN coefficients ar(f), r = 1, 2, ..., and these coefficients are given by
ar (f) _
(p(r))-1
Tr dr
Md(f), r = 1,
2, ...
(2) and, conversely, the existence of all ar(f), r = 1, the existence of all Md(f), and
M d (f)
rTd p(r)- a
= d 1
,
2, ...,
implies
r(f).
Proof. (1) Is obvious: M(crf) = Edir d µ(r/d)- Md(f). (2) For every x > 0, rid nx
cr(n) = 2 f(n)
f(n)
wax
_
tTd
t
nsx
r Td tjn,r) f(n)
n.0 mod t
t µ(r/t)
al(d/t)
µ(s) = d
So Md(f) exists and the formula given is valid.
I
nsx
n-0 mod d
f(n).
VIII.4. Formulae for Ramanujan Coefficients
281
So, for the calculation of RAMANLUAN coefficients, it is crucial to
obtain the mean-values Md(f). In order to be able to calculate these mean-values, we begin with the definition M(d)(f) =
x
lim
M
x-1
nsx, gcd n,d) = 1
if this limit exists. proposition 4.2. Assume that f is a multiplicative function, for which a) the mean-value M(f) exists and Is non-zero, b) for any prime p the series Y-kkO P-k.If(pk)I Is convergent,
c) and all the mean-values M(d)(f) exist. Then, for every prime p, 1+p and for every Integer d,
(4.1)
M(d)(f) = M(f)
p IT
(1 + p1. f(p) + P-2. f(p2) +
l
... 1-1
Proof. According to Proposition 3.2, M(f)
1+
C-1(o)
n=1 n-O f(n),
and similarly for M(d)(f) = M(f Xr), where Xd(n) =
I
If gcd(n,d) =
1,
and = 0 otherwise. Using the multiplicativity of f and of f Xr, and noting
2.f(p2)+... ), we obtain M(d)(f)
pI a ( 1 + p 1. f(p) + P-2. f(p2) + ...) = M(f).
Thus M(d)(f) t 0 and ( is true.
1
+
P-1.
f(p) + P-2. f(p2)
+
...
) $ 0, and (4.1)
We remark that for a function f in D1 the mean-values M(f), Md(f) and M(d)(f) do exist (see Chapter VI, Section 1). Moreover, if f Al and M(f) * 0, In the proof of VII, Theorem 5.1, the convergence of c
S3,1 (f) has been shown; hence all the series Y_ kiop-k If(pk)I are convergent. If 11 f 11 < co for some q > 1, then If(pk)1 s C. pk,'9, therefore a
Ramanujan Expansions
282
the series I + p-1 f(p) + P-2. f(p2) + we obtain the following proposition.
...
is absolutely convergent. Thus
Proposition 4.3. If f Al Is multiplicative, with mean-value M(f) * 0, then all the mean-values M(d)(f) exist and formula (4.1) holds. e
Theorem 4.4. Assume that f f Al Is multiplicative, and that M(f) * 0, Then the mean-values Md(f) and the RAMANrIIAN coefficients a(f) exist, and the maps
d '- Md(f) / M(f), and r '- ar(f) / M(f) are multiplicative. There are product representations for Md(f) and ar(f) as follows: Md (f) = M(f)
f
8+1
lf(p8)
TT
+
f(pps+1 )
p
Ps
13811d
2 f(p)
+
P
+...
1-1 I
,
and M(f)
ar(f) _ 9(r)
= M(f)
n \k
m
pSllr
f
f(pk+s-1)11, (1 +
((
+
p
O
fZ
+...
P-k, f(pk)
rT ( kx6 E
I-1
Y1
kzp
pShIr
Proof. For a fixed Integer d = p1'
write m = pi'
pr
...
pr'
$,
where gcd(&d) = 1. Then Md(f) = =
lim
°-+ 1+
d-1.
E
c 1(a) lim
n=O mod d
n-O f(n)
-1(a) E m-° , f(m , d )
0-+ 1+
m f(p;`'+S,)
=
lim ° -3 1+
1(
)' Et
(t,d)=1
f(Z)
$°
E µx0
E µ?O
By VII, Theorem 5.1, the multi-series E
µ, zo
µ °+S , P,
...E µr zO
...
112:0
µ+S, f( p,
.,µ,+S
) .
... f( p,µ,+S
.,µ,+S.
M
fpµ.+s,
... P 0+
Is absolutely con-
vergent in a 2 1. So we obtain (from (4.1))
M (f) = µE20 ...E
....
(d) (f)
VIII.4. Formulae for the Ramanujan Coefficients
8 = M(f) n f(s)
+
f(ps+t) p
p
p8Il d
+
...
283
1 + f(p)
2
)-1
+
p
p
This proves the formula for Md(f) and the multiplicativity of the map d H Md(f)/M(f). Proposition 4.1 yields the fact that the function
r - ar(f)/M(f) _
r
(f)
{(P(r))-1
dr d
U
Md
is multiplicative, and
as(f) M(f)
I
_
y(p)
=k
S
k=o P
f(pk+8-1))) CI +
P-k' (f(Pk+s) -
k' ( f(Pk) -
P
f(pS-1))
Corollary 4.5. Assume that f
e
f(p)
+
f
)
p
+ ...
-1
. lk o P-k. f(pk) )-1 At Is
strongly multiplicative, and
M(f)*0. Then Md(f)
=
a(f) r = M(f) Proof. M(f) = lim IT ( 1 °->1+
P
f(d) d
r 1+
g(r) +
f(p)-1 )-1
. nPirI` 1 -
1+
f(p)
I. )
P-6-MO-1)), therefore
(I
+
f(a)-1 ) P
1
Is non-zero for every prime p. Theorem 4.4 implies the above formulae.
Corollary 4.6. If f e Al is a completely multiplicative arithmetical function, for which the mean-value M(f) is non-zero, then Md(f) = M(f) d-1 f(d), M(f)
ar(f)
Tr f(d) µ(r/d) = M(f) (p(r))-1 (µ*f)(r) dr
Corollary 4.7. If f e A 1 Is 2-multiplicative and M(f) * 0, then MPs (f) = M(f) f(p) { p + f(p) }-1 If S = 1, and MPs (f) = 0 If S 22, aps (f) = a
P
f) (-1 + f(P)/(1+ P 'f(P))) If S = 1,
S (f) = - M(f) .
{(P(p2))-1
ap8 (f) = 0 If S Z 3,
( f(P)/(1 + p-if(P)) ) if 8 = 2.
Ramanujan Expansion,
284
Finally, we give some formulae containing the ERATOSTHENES trans_ form. The WINTNER condition (2.1) Implies Ensx If'(n)l = o(x). Then
Itln f'(t)
2:nsx,n-0 mod d f(n) = 2:n-.,n--O mod d
t-1. NO . ((x/d)
= ItSx
so Md(f) =
t
a
=x
t
' ztsx t
d-1
gcd(d,t) + 0(1) ) gcd(d,t) + o(x);
1.f'(t)
gcd(d,t) exists and the following result holds.
Theorem 4.8. If the series
is absolutely convergent, then the mean-values Md(f) and the RAMANIIIAN coefficients ad( f) exist for d = 1, 2, ... . If f is multiplicative in addition, then the meanvalues M(f) and Md(f) are given by (2.1)
M(f) = IT (1 + p-1 f'(P) + P-2' f'(P2) + ... P
f'(P)+ f p2 p p
),
2
Md(f) =
(1 + p7d
+
x n (1 + V(p) +
... ) ...
p'Ild
+ f'(Pk) + f (a" )+ f '(P'*') ) p
If f is multiplicative and M(f) $ 0, then Md(f) f (p') M(f)
P`Ild
ad(f)
M(f)
+ ... + f.(Pk)
(1 + f'(p) + ... +
=
P`lld
+...
f.(pk-1) +(I+ +
f (p)
f-)L
p
P
1
+
f'(
+
... ).
P
+
f (p2))
+
-1
P)lf.(Pk) + f (p ) +...)) + ...
and the maps d H Md(f)/M(f), r H ar(f)/M(f) are multiplicative.
VIII.S. POINTWISE CONVERGENCE OF RAMANUUAN EXPANSIONS
A large class of arithmetical functions f, for which the RAMANUJAN expansion (1.1) is pointwise convergent, is the set of multiplicative functions in 442. This is a consequence of the main Theorem S.1 in Chapter VII.
VIII.S Pointwise Convergence of Ramanujan Expansions
28S
'T'heorem 5.1. Assume that f Is a multiplicative function in A2 with mean-
value M(f) * 0. Then its RAMANUJAN expansion is polntwlse convergent and
Z a (f) r=1 r
cr(n) = f(n) for any n e N.
Remark 1. In general, convergence is neither absolute nor uniform In n.
Lemma 5.2. If f is a multiplicative function In
A2
with mean-value M(f) * 0, then - denoting by a*(f) = {M(f)}-1 ar(f) the normed RAMANUJAN coefficients - the following two series are convergent: ap,
(5.1)
P
(5.2)
P
Remark 2. The same proof (with a slight modification in (2)) works, if f e s4q for some q > 1 is assumed. Proof. (1) VII, Theorem 5.1 yields f e 92, and so, in particular, the series
eP = ( 1-p 1) . z P-k. f(Pk) k=O Is convergent for any prime p. II f 11q < oo implies If(n)I s c ni, and so there exists a prime p1 with the property I Zk21 P-k f(pk) I s 2 for all primes p 2 p1. Therefore, IePI z ;. According to Theorem 4.4, we obtain (for every p z p1) .
- ap = (p
k=O
1- f(p)
=
. Y P-k. (f(pk) - f(pk-1) )
eP}-1
+
1
eP
p
( (
1- f(P) )2 p
1
+
eP
1
(
1- f(p) + 1) / k21 p
f(pk)-f(pk-1).
pk.'
Summed over p Z p1, the three series on the right-hand side are convergent:
1-f(p) P
P
Z P
1
(
eP
l
I
= S1(f),
I
P
1-f(P) +1), E l p
e
(
1
P
-12 )
f(P))21 = 0(1) + o(2]1 f (P) P
f(Pk) f(Pk-') 1
pk"
This proves the convergence of (5.1).
P
P
C7(S2 (f) + S 3,2 (f)) + 0(1).
286
Ramanujan Expansions
(2) Choose a prime p2 such that p-1 If(p)I s 2 for every p II
P
I-1 s 4 for p ;t max{p1, p2}, where IP = (1 - p-1. f(p) )
(5.3)
k
O
p2' Then
P-k. f(pk)
Thus aP =
(
kO 1
IP
P-k,
f(pk))-1
k:1
(f(P) -1 `
f(P2) - f2(p)
+
p-1 f(p2)
P
- f2(P) I
1
1
f(pk) - f(pk_')f(P) )
+
k:3
p2
1 I+I
= °(I p-1
P-k. f(pk) -
pk
+ p-3/2 ).
p2
This gives
(5.4) p IaP*I2 = 0(P-1. If(p) -112 + P-3' If(P2)I2 + P-3. If(P)I4 +
P-2
and so P p Ia P *12 < oo, estimating the sums over the terms in (5.4) by C')(S2(f)), O(S3 2(f)), 0(S2(f)) and 0(1) respectively. Proof of Theorem 5.1. (1) We first prove the convergence of the RAMANUJAN expansion at the point n:
E a -c (n)rsx = Er digcd(r,n) aE
rsx r r
dT.
d- rsx,r-OE mod d a-r µ(r/d),
Thus, in order to prove the convergence of rE arc r(n), it is sufficient to show the convergence of the series E a* . µ(r) for every d. Write r d = n ps = t D, where t = ri PS Id,PS=O p8, and denote the squarefree kernel of t by a(t) = TPIt p. Then
E a* µ(r) = µ(«(t)) a"(t) rsx/a(t gcd(r,t)=1 P(r).arD
rs x
*
aD
rsx
t)
r -1 u(r),
where
µ(r) r- arD/aD,
if gcd(r, t) = 1,
0,
otherwise.
u(r) _
yIII.S Pointwise Convergence of Ramanujan Expansions
287
The convergence of 7- r 1 u(r) remains to be proved. Since x
-1
' %sx ( 1*u)(n) = Zrsx r 1'u(r) + of x-1 ' rsx lu(r)I),
it suffices to show that (a) M(1*u) exists, (b) M(Iul) = 0.
Proof of (a). The 2-multiplicative function T = 1* u belongs to 92. Since p-1'
p-' ' ( 1 - T(P))
/a*
u(p) = a*
= ap
for every p,' d, the series S 1(T) and S2(T) are convergent (see Lemma 5.2), and
S3 2(T) = P k:2 P-k' I f(pk)I2 =
{p(p_1)}-1
p
.IT(p)12
= o(P Ia*12) = 0(1).
VII, Theorem 5.1 Implies T e .p42, and so M(T) exists.
Proof of (b). By partial summation the estimate E
rsx
r-' - Iu(r)I2 5 IT (1 psx
+
Iu(P)I
) s exp { E
psx
P
I1-pf(P)I2 )
0(exp
Iu(P)I
}
P
= 0(exp S2(f)) = 0(1)
P implies x-1' I lu(r)I2 = o(1), and (b) and the first part are proved. rs x
(2) For any fixed n, the DIRICHLET series
A(a) = Er:l r-O' arCr(n)
is convergent for any a > 0. In fact, it is absolutely convergent since p-ko,laP. cp.(n) r-' ,ar cr(n)I s P:cx IT rZ kYo I
and, because of cp.(n) = -1 if k = I and p,j' n, and cp(n) = 0 if k 2 2 and the product Is absolutely convergent:
(E P-o'Ia*I )2 s Z P. la*I2 . Z P-(1+20) < P
P
P
P
P
CO
Ramanujan Expansiogs
288
(by Lemma 5.2). In o > 0 the DIRICHLET series A(d) has the product representation
E r-° a*c r r (n) = IT b p(o),
A(o)= M(f)
r>1
P
with factors
k aP. bp(d) = ep ko p-o
c p.(n)
According to the continuity theorem for DIRICHLET series, it suffices
to show that:
(c)
k
if p8IIn, then ep
(d)
O
ap. cp.(n) = f(ps),
IT b (o) = 1. P
lim
c-4O+ p>n
Proof of (c). It is easy to show (see Exercise 3) that 1
cp - P
oSkss+1 cp.)(n)
Ps. ifpslln,
-t0
otherwise.
Therefore, 1-
1) p
E
Osks8
1
M( f c P ) k
p
M(
P
M(f) - MP(f)).
s+1) =
So we obtain bp(0) = ep kkO
*
OsksS apk
.
* k S Y(p) - aps+i' P
f p) M(f) \ `1 - P) o kSs M`f Cpk) fpp) k
f(Ps)
1 k:O
(
1-
M (f) =
M(f)
p
M(f cps+i)l
f(ps),
Proof of W. cpk (n) = -1 [resp. 01. if k = 1 [resp. k i 21 and p
> n.
Therefore,
b (o) = e P
P
P
S (o), P
where 8 (o) _ (1- p-') ( ep - 1). The relation P-k
ep - 1 =k 1
(f(Pk) -
f(pk-1))
=
P (f(P) - 1) + p-2 f(p2) + C7(p3/2)
VIII.6. Still another Proof for Parseval's Equation
289
shows that the following series are uniformly convergent in 0 s 0 s 1: 1-p-o) P
P
P
P
(f(P)-I
C)( E
P-21f(p2)1 + p-3/21
P
Z18 (0)12=C)(Z1' (f(P)-1)12+1)=0(1). P
P
P
P
Therefore, 11 b P(o) is uniformly convergent In 0 s 0 s 1, and p>n
11b P(0)= ITb (0) = 1. p>n P
lim
o->O+ p>n
This concludes the proof of Theorem 5.1.
VIII.6. STILL ANOTHER PROOF FOR PARSEVAL'S EQUATION
In Chapter VI two proofs for PARSEVAL's equation were presented for functions f in 22. In this section, in the special case where f is multiplicative in addition (and M(f) $ 0), a third proof is given.
Theorem 6.1. Assume that f e A2 Is multiplicative, with mean-value M(f) * 0 . Then PARSEVAL's equation E
1sr 0. Moreover, BESSEL'S Inequality yields the I
convergence of the series
Y_
lsr n, and kp are even functions. Calculate the mean-value of in Ikpl2
Ikpl2
and
two
different ways.
xi2: Ik(n)I2=Ie PI2 nsx P The last sum
nsx
Osk,esl+logx/loBp
a*,a ex-1znsxck(n)ce(N). P P P P
cP,(n) c Pe(n) equals x 9(pk) + 0(p2k), if k = E, and
it is 0(pk+e), if k * t. And
a*, = Pe-1m2, ik f(P-)-f(p= o(P'k). p'" P Therefore,
x 1nsx 2 IkP(n) 12 = le P I2 . Osksl+logx/loBpI a*,P 12. Y(Pk)
+ leP 12x1 Osk,esl+logx/logy C7(1). This equation implies M(lkp12) = lep12
k2:0
I apk 12 cP(Pk)
On the other hand, kP(n) = f(pk) if pklln (see (c), p.288) and so
x' nsx IkP(n)12 = Osksl+logx/loge P
k. If(Pk)I2 pk
msx/p`,p.l' m
hence,
M(Ikp12)=(1-p
).
kOp-k,lf(Pk)12.
Comparing both representations of M(lkp12 ), (6.1) is proved.
VIII.B Exercises
291
VIII.7. ADDITIVE FUNCTIONS
A. HILDEBRAND and the second author of this book [1980] proved the
following result for additive functions. We do not prove this here, but refer instead to the paper quoted in the bibliography. Another proof for this result, in sharpened form, was given independently by K.-H. INDLEKOFER
Theorem 7.1. Assume that g Is an additive arithmetical function. If q z 1, then the following three conditions are equivalent: (i)
g E
(ii)
The mean-value M(g) exists and II gliq < oo.
(iii)
The following three series are convergent:
uI
.
1'g(P)
E
P-1.
Ig(P)I51
Ig(P) Isl
and
z
p. k21, Ig(p )I>1
p k'lg(pk)Iq.
VIII.8. EXERCISES
1) Let f ' = µ * f be the ERATOSTHENES transform of the arithmetical function f. If if'(m) f'(n) I Xm21 Zn21 lcm[m,n]
o
x-
c
nsx,n-0 mod d r
Iw(r)
if rid,
d 0,
if r.}' d.
Ramanujan Expansions
292
3) (a)
Prove thatrd cr(n) = d, if din, and
a cr(n) = 0, id d4' n.
r
(b) For all Integers n z 1 and 8 Z 0, prove that
l :!E:: 0sks8
-
p-1
Osks8+1 Cp)(n)
CP
p6 , If ps ll n,
0
otherwise.
4) Verify the calculation of the RAMANL[IAN coefficients and the point-
wise convergence of the RAMANUJAN expansions for the arithmetical functions f = o/id, and f = 9/id, given in VIII.2, p.274.
S) Let f be a multiplicative arithmetical function; denote the ERATOSTHENES transform by f' = f *V. Prove that Iif' 112 < 00, whenever Of 112 < co is true.
6) Assume that f e At is multiplicative, and M(f) * 0. Prove that for all primes p, for which If(p)I < p, the formula f(p) _
1
M(f)
p-1
1 f() + E p k. (f(pk) k2e
pt-1
ape j
i - LL)} p
kyo p-k f(pk) 2:
holds. Hint: use Theorem 4.4.
7) If v is the function used In the proof of Theorem S.1, show that
M(1*v)=pktn(
1+v(p) p
293
Chapter IX
Mean- Value Theorems and Multiplicative Functions, II
Abstract. This chapter Is a continuation of Chapter II. We are going to give proofs for two, deep mean-value theorems for multiplicative functions, namely one due to E. WIRSING [1967], with a proof by A. HILDEBRAND [1986], and the other due to G. HALAsz [1968], with an elementary proof given by H. DABOUssI and K.-H. INDLEKOFER [1992]. This proof uses ideas from DABOUSSIs elementary proof of the prime number theorem. HILDEBRANDs proof uses a version of the prime number theorem with a [weak] error term, and thus, while HILDEBRANDs proof
does not give a new elementary proof of the prime number theorem, the DABOUSSI-INDLEKOFER proof does.
Mean-Value Theorems and Multiplicative Functions,
294
II
IX.1. ON WIRSING'S MEAN-VALUE THEOREM
The mean-value theorem due to EDUARD WIRSING for real-valued func-
tions has already been mentioned in II.S. In this section we restrict ourselves to real-valued arithmetical functions f of modulus IfI s 1, and we give A. HILDEBRAND'S proof [19861 for the following theorem. Theorem 1.1 (E. WIRSING, 1967). For any multiplicative, real-valued arith-
metical function f satisfying IfI s 1, the mean-value M(f) = lim
(1.1)
x-1
X
f(n)
exists. If the series P p 1
(1.2)
(1 - f(p))
is divergent, then the mean-value M(f) Is zero. Corollary 1.2 (ERDOS-WINTNER Conjecture). Any multiplicative arithmetical function assuming only values from the set {-1, 0, 1) has a mean-value.
Corollary 1.3 (Prime Number Theorem). The MOBIUS function n H µ(n) has a mean-value.
Remark. In fact, the Prime Number Theorem R(x) - io x x , as x --) co g
follows from Corollary 1.3. However, the proof of Theorem 1.1 (in the
stronger version of Theorem 1.4) and of its corollary uses a stronger version of the Prime Number Theorem, and so this result cannot be considered to give a new proof of the Prime Number Theorem. Corollary 1.2 is obviously a special case of Theorem 1.1, and the assertion of Corollary 1.3 for the MoBIUS function is contained in Corollary 1.2.
The divergence of I P
p-1
(1 - µ(p)) _ Z 2 p-1 implies M(µ) = 0. The p
deduction of the Prime Number Theorem (1.3) in the equivalent form (1.4)
fi(x) = Z A(n) - x nsx
IX,1. On Wirsing's Mean-Value Theorem
29S
possible by elementary ( though somewhat tricky) arguments, as shown by E. LANDAU. We start with the arithmetical function
is
h=log-t+2L',
(1.S)
where e is EULER'S constant, and t = 1 * is the divisor function; s denotes the unit of the ring of arithmetical functions with convolution, and from Chapter I we know the convolution relations 1
A = µ * log, l
=
i * t , E = it * 1.
Therefore,
nsx
nsx din
= dEmsx X µ(d)
h (m).
On the other hand,
{A(n) - 1 + 2 L° E(n)} = cp(x) - Ix] + 2 L°,
and so the Prime Number Theorem (1.4) is proved as soon as
IZ
µ(d)
h (m) = o(x) (as x - co)
'
is proved. A DIRICHLET summation (see
2),
1,
using the summatory
functions
M(x) = Z µ(n), and H(x) = Z h(n), nsx
nsx
gives, with some parameter B = B(x),
E Z 1i(d)
h (m) =
I
dsx/B
=d x/B
µ(d)
Z
msx/d
h(m) +
ii(d) H(d) +
Z h(m).
msB
B
Z
x/B 2, the inequality
P-I.I
Z . x nsx,pln
psx
w n
-1x .nsx Zw n
x
Z
Iwn12
nsx
holds.
See I, Exercise 16.
Lemma 1.6. Assume that f Is a non-negative multiplicative arithmetical function, satisfying the two conditions
I f(Pk)
p" s x
log pk 5 Y1 x,
Z Z p-k f(pk) 5 Y2 psx kk2
with some constants Y1 > 0, Y2 > 0. Then, with some constant
Y,
depending only on Y1 and y2, the estimate x-1
21
nsx
f(n) 5 y- exp (psx Z P -1 (f(p)-1) )
holds.
Proof. This result may be deduced from II, Theorem 3.1 (3.3).
O
Examples. Lemma 1.6 can be applied to multiplicative functions f satis-
fying 0 5 f
5
1,
or 0 5
f(pk)
5 k+1 for all prime-powers pk, or
298
Mean-Value Theorems and Multiplicative Functions,
0 s f(pk) s al
2
for all prime powers
,
pk,
II
with some constants
X1 > 0, 2 > >2 > 0.
Lemma 1.7. Uniformly in x Z 2, and for all real-valued multiplicative arithmetical functions f satisfying -1 s f s +1, the estimate
Z n_'
-
P
x
holds.
Proof. The function g = I * f is multiplicative and satisfies Ig(p)) 1+ f(P)I = + f(P), and Ig(Pk)I s I + If(P)I + ... + If(Pk)I s k+l. There1
fore Lemma 1.6 implies x-1
-
12: g(n) I nsx
s x-1
E
nsx
I g(n)I
Y3 exp( Z p-1' (Ig(P)I - 1) )= p
Z psx
p-1
f(p)).
This estimate, together with
I n-1 f(n) nsx
=
1x nsx f(n).[ xn ] + O = 1x nsx g(n) + O(1),
where IOI s 1, gives the assertion of Lemma 1.7.
11
IX.2. PROOF OF THEOREM 1.4.
In Chapter II, the summatory function of f was denoted by M(f,x) (= nsx f(n)). Let us define the function A(f,x) by (2.1)
A(f,x) = I . M(f,x) = I . I f(n). nsx
Next, we use the (2.2)
and
(2.3)
notation
S(x)
S(x) = E psx
f(P)),
S(x) = min ;, a- SW
1x.2. Proof of Theorem 1.4
299
The proof of Theorem 1.4 depends on an oscillation property of J1l(f,x), stated in the following lemma. lemma 2.1. With an absolute constant C, for all multiplicative functions f, satisfying -1 s f s 1, with divergent series (1.8), the "oscillation condition" (2.4)
I
ftt(f,Y) - Jl1(f,x) 15 C
'
( log
log x
loYx)
)
holds in 3 s x s y s x5/4
proof of Theorem 1.4. We use the notation S(x) and S(x), introduced at the beginning of this section. First, with positive [absolute] constants Yt, 1 s I s 4,
S(x) Z Yt ' exp (- ;
(2.5)
P
p-1)
YZ '
( log x)-3
and, if 3sxsysx 1+S(x) (2.6)
log
Y3
log x )-i S ( log S-1(X) )-' s Y4 log(2y/x)
S(x) + 1)-1
21
Applying Lemma 2.1 and inequality (2.6) we obtain, in 3 s x s y s A(f, X) = A (f, Y) + O((S(x) +
x1+s(x)
1)-,).
An integration of this equation gives a representation for AZ(f,x) "in the mean",
m(f,x) _ (S(x) log
Y-
x)-1 '
Jx
I1(f,Y) dy + O((S(x) +1))
The integral is x1+S(x)
(x1+S(x)
Jx
Y-2. Z f(n) dy
Y-2
f(n)
max(x,n)
f(n)
(
n:9 x 1+S(x)
dy {max(x,n)}-1
-
x-1-S(x)
ns c1+S(x)
= E x< nsx i+s(x)
f (n) n
+ JIi(f, x) - Ii(f, x1+s(x))
_ 1 x<nsx1+s(x) f (n + 0(1),
and we obtain fti(f,x) = (S(x) log
x)-1 . C
> x (log x)-', and S(X,y) Z (log
(2.8)
X)-3/2
Then R(x,y) is of the order 1
1 Is always assumed. For every real t, as o -4 1+,
E p-a (1 - Re
(3.15)
J
P
converges monotonically increasing to infinity. The divergence of (3.15)
is uniform on every Intervall [-K,K]. This can be seen from a variant of DINI's theorem [see P. D. T. A. ELLIOTT 119791, Lemma 6.7, p.241, and the remark on p.242) ] or directly in the following way. Define p-a (1-
GCS(t) _
)), o > 1,
Re I f( P
and assume that divergence of (3.15) is not uniform. Then there exists a constant c > 0, a sequence 0I > d 2 > ... > 1, 0n -) 1, and points to e [ - K, K ] such that Ga (tn) s c-1 for every positive integer n. Taking a suitable subsequence we may assume to --> to. Fix o e ] 1, 00[ . There exists an integer n0 such that on s e for every n 2 n0, and so Ga(tn) 5 G0 (tn) 5 c 1.
Hence Ga(t0) s c-1 in o 11,0 0 1. This contradicts the divergence of (3.15) at t = t0, if d - 1+. e
Using the PARSEVAL equation (3.12), and then (3.9) (with 8 = 2, c = 2), we obtain (3.16)
f°
I
F99) I2dt
(0-1)-1
= >
C, then there Is a continuous extension
F: X -4 C with compact support, Fly = f, vanishing outside U.
A.2. ELEMENTARY THEORY OF HILBERT SPACE
Let X be a HILBERT-space with inner product
,
>
,.2. Elementary Theory of Hilbert Space
for each e
317
El. The FOURTER coefficient of x E X with respect to e E E is denoted by 11 a 11
=
1
E
z (e) = <x,e>.
An orthonormal set E is "complete" if <x,e> = 0 for every e e E Implies 0. For example, the set of functions x H exp(27ti nx), -2, -1, 0, 1, 2, ..., is complete in L2([0,1], A), where A is the LEBESGUE measure.
The GRAM-SCHMIDT orthonormalization process permits us to construct from any at most countable set E* _ { et e2*, ... } of linearly independ-
ent elements of X an orthonormal set E tional property LinC{ et
,
..., en } = LinC{ et
,
..., en
} with the addi-
et
,
}
for n = 1,
e2
,
...
2, ...
.
If X (0) contains a dense, countable, linearly independent set, then the GRAM-SCHMIDT orthonormalization process leads to a complete orthonormal set E in X. BESSEL'S Inequality. Given an orthonormal set E $ 0 In an Inner-product space X. Then E eeE
I
<x,e> 12 S 11x112,
and so the set { e e E, z (e) t 0 } Is countable.
Theorem A.2 1. Assume that X Is a HILBERT space, and E c X an orthonormal set. Then the following properties are equivalent: (i)
E Is complete.
(ii)
The smallest linear subspace of X containing E Is dense in X.
(iii) For every x E X, PARSEVALS equation is true: 11x112=E I<x,e>I2. eeE
The series contains at most countably many non-zero summands. (iv) For every x, y the [generalized] PARSEVAL equation holds: <x,y> = Z <x,e> eeE
'
.
Appendix
318
(v)
Every x e X has a FouRIER series
x-Z
ecE
with at most countably many non-zero coefficients <x,e> [the corresponding e's are denoted by e1, e2, ... ], and N
llm oo
IIx - nsN E
Theorem A.2.2. Let F be a bounded linear functional F: X - C, defined on the HILBERT space X. Then there exists a unique element y e X "representing" F: F(x) = <x,y> for every x e X.
Moreover, the operator norm
IIFII =
sup
Ilxll=l
IF(x)I
equals
Ilyll.
Theorem A.2.3. Assume that X Is a HILBERT space. The set -8(X) of all bounded linear operators T : X - X is [with composition as multiplication] a BANACH algebra with unit element. For every T e 8(X) there Is a unique "adjoint" T5 e 73(X), defined by = <x,T*(y)> for every x,y e X.
Moreover, T** = T, and [the operator norm J
IIT*II = IITII.
Some properties of the adjoint operator are listed below: (i)
(T1 + T2)'e = T1*+ T2*
(ii)
(aT) =aT
(iii)
( T1 ' T2) * = T2*
(iv)
IIT*
(v)
If = 0 for all x,y e X, then T = 0
(vi)
If = 0 for every x e X, then T = 0.
^
T1*
TII = IITII2
319
A.3. Integration
A. 3. INTEGRATION
reference
Our standard
are
the books by W. RUDIN
[19661
and
HEWITT-STROMBERG [1965].
Given a measure space (X, 4, µ), where 4 is a o-algebra of subsets of
x, and It is a non-negative measure, a function f : X ---3 C is termed measurable if f-1(O) is in 4 for every open set 0 in C. A [measurable] simple function is a finite linear combination of characteristic functions of [measurable] sets. The integral of a measurable simple function [ over
a set E in 41 can be defined in an obvious way as a finite sum. If f: X --j [0, ool is measurable, its integral [over E E 4 ] is defined as
I
E
f dµ =
sup
Osssf, s simple, measurable
fE s
dµ.
The extension of this definition to functions f: X - C is done via linearity.
A property is said to "hold almost- everywhere" if the set of points, where it does not hold, has measure zero. Concerning countable limit operations, there are important convergence
theorems:
Lebesgue's Monotone Convergence Theorem. Assume that {fn} Is a sequence of measurable functions fn: X - IR, satisfying 0 s f1(x) s f2(x) s ... s oo and limn fn (x) = f(x) for almost every x c X. Then f Is measurable, and lim
n --> m
fx
fn dµ = fX f dµ.
Fatou's Lemma. If f n : X -4 [ 0, oo ] is a sequence of measurable functions, then
fX lim inf f n-) m n
d1i s lim inf f
n-4 m X fn
dµ.
Lebesgue's Dominated Convergence Theorem. Let {fn } be a sequence of
complex-valued measurable functions defined a.e. on X with the property
320
Appendix
Y
fX
n=1
'fnI dµ < oo.
(This is equivalent to f X Yn=1 1fnI dµ < co.) Then the series
1,
=1
fn(x) = f(x)
is convergent for almost all x E X to a measurable function f, and
fX Ln=1 fn dµ = Zn=1 fX fn dµ. A useful discrete version of the Dominated Convergence Theorem is the following result.
Corollary. Let a sequence fN: IN - C, N =
1, 2, ..., of arithmetical functions be given, and let F: N -3 [0, co] be a function satisfying
Z n-1 F(n) < co. If IfNI s
F for every N E IN,
and If the pointwise limits limN , m fN(n) exist for every Integer n, then the limit lim N-) co Z°°n=1 fN (n) exists, and
Nlim co
Z
n=1
fN(n) = Z
n=1 N >
fN(n).
An important tool for the construction of integrals is the Theorem of F. Rmsz. Let A be a monotonic lx s y A(x) s A(y)) linear functional on the vector-space of continuous functions on the locally compact HAUsDORFF space X with compact support. Then there Is - a e-algebra f4 of subsets of X, containing all the BORBL sets of X, and - a [positiveI measure µ on A such that A(f) = f X f dµ
for every continuous function f on X with compact support. In addition, the measure has the following properties: (a) µ(K) < co for compact sets K. (b) µ Is outer regular, which means µ(E) = inf µ(V) for VDE, V open any E In A
A. 4.
Tauberian Theorems
321
(c) µ Is Inner regular, which means µ(E) =
sup
KcE, K compact
µ(K) for
E in 14
if E is open or if E has finite measure or if X Is o-compact (X is a countable union of compact sets) and HAUSDORFF.
(d) µ Is complete: a subset of a set E In 4 with measure zero is In 4 again and has measure zero. Theorem of FuBINI. Let (X, 4, µ) and (Y, 73, v) be a-finite (a countable union of finite measurable sets) measure spaces. If f Is a complexvalued (A x 73)-measurable function, defined (a.e.) on X x Y, which satisfies
fxxY
Ifl d(µ x v) < co
or f x(fYIfI dv)dµ < . or fY (f
Ifl dv)dv < co,
then
f xxY f d(µ x v) = f x(f Y f dv)dµ = f Y(fx The product-a-algebra 4 x
("3
f dµ)dv
.
is generated by the measurable rectangles
ExF,where B 4,Fe73.
A.4. TAUBERIAN THEOREMS (HARDY-LITTLEWOOD-KARAMATA, LANDAU- IKEHARA)
A good reference for the topics dealt with in this section is WIDDER [1946]. The HARDY- LITTLEWOOD-KARAMATA Tauberian Theorem will
be formulated in a version for LAPLACE-Integrals, from which a version
for DIRICHLET series, as well as for power series, will be deduced. We need the notion of a slowly oscillating function L: [a,oo[ -)1R.
L Is called slowly oscillating If L Is continuous, positive, and satisfies lim
x -) m
L(cx) / L(x) = 1
Appendix
322
for every c in 0 < c < co.
For example, the functions x H (log x)k, x H loglog x, x H exp(4og ) are slowly oscillating. Theorem A.4.1 (HARDY-LITTLEWOOD-KARAMATA). Assume that A(,)
Is a real-valued, non-decreasing function defined on the Interval Moo I, and A(0) = 0. Let the LAPLAcE-Integral
2(o) = a
f-
°LL du
O
be convergent for any o > 0, and suppose that for some slowly oscillating function L and some t > 0 the relation
f(c)
lim
o -3 0+
o--C
{
L(0-1) }-1 = y
holds (where L(o-1) / co In case that t = 0). Then, as x -> co,
A(x) -
xt
I'(t+I)
L(x).
For a proof, see, for example, HARDY [1949], p.166, or SCHWARZ [1969].
Partial integration gives n=1
an
n-6
=-
f
Y_
n - 1, then the series Y_ log l l + an ) converges absolutely if and only if the series n a n Is absolutely convergent. If Re an > 0, then the product ff (1 + an) Is absolutely convergent if and only if the series X an is absolutely convergent. n
Finally, we give a result on infinite products which is useful in number theory.
Theorem A.7.1. Assume that the two series p p
1a
,
P
Z P-2. la 12 P
P
where p runs over the primes (in ascending order), and where the a are complex numbers, are convergent. Let P
a yg(p,o): [1,1+S]-aC be a continuous functions satisfying bP < oo.
Ig(P, 6)I 5 bp, and Then
P a
(a) the product TI (i + P + g(p, a)) Is convergent for every 6
e [1,1+8], and,
IT 0+ a + g(p, d) )
(b) if it is supposed in addition, that A = lim
o-1+ P
exists, then (
p
a
P
Proof. In Izi s 2, write I + z = exp (z + R(z) ), where R(z) = log( I + z) - z = O(Iz12). Choose a prime po so large that I p-1 a + b s 2 for every P P p0. Then p I
A.B. The Large Sieve
329
g(p,(j)I s; for p z po, and o f [1,1+8]. a+ P Then, for p1 z p 0 and o E [1,1+8],
II PosPSP1
a
(1 +
a + g(P, o) p =
exp{
Pos
The convergence of in o Z 1, and lim°--* theorem
imply
( SP
a
l
/a
f
l
g(p, °))} P + g(p, e)) 1 exp { PosPSPiRl o+ p
implies uniform convergence of Z p-O a P P E p-O aP = p-1 aP. The assumptions of the P P p-1. a
P 1+
the
P
uniform
convergence
of
P
g(p,o)
and
R(p-O aP + g(p,o)) in t s 0 s 1 + 8, and we obtain the assertion (a) by letting p1 tend to infinity and o to 1+. (b) Is then obvious. P
A.B. THE LARGE SIEVE
References for this section are, for example, E. BOMBIERI, Le grand crible dans la theorle analytique des nombres, asterisque 18 (1974), H. DAVENPORT [1967], H. HALBERSTAM & K. F. ROTH [19661, M. N. HuxLEY [19721, H. L. MONTGOMERY [1971], H. L. MONTGOMERY & R. C. VAUGHAN, The Large Sieve, Mathematika 20, 119-135 (1973), H.-E.
RICHERT, Sieve Methods, Bombay 1976, W. SCHWARZ, Elnfiihrung in Slebmethoden der analytischen Zahlentheorie, Bibl. Inst. (1974).
We only need one aspect of the "Large Sieve", namely an estimate of an exponential sum S(x) _
a
exp(2ni n x)
M<nsM+N n In the mean, taken over well-spaced points.
Theorem A.B.I. Let x1, x2, ... xR, where R z 2, be real numbers, distinct
modulo one. Put (A.8.1)
8 = min III x r r*s
- xg
III
,
Appendix
330
where III
. III
denotes the distance to the nearest Integer. Then
E 1srsR
I S(xr)12 s (n N + 8-i)
E Ian 12. M <ns M+N
The expression (7c N + 8-1) on the right-hand side may be replaced by (N + 8-1), as was shown by H. L. MONTGOMERY and R. C. VAUGHAN.
A rather simple proof of theorem 8.1 may be obtained using GAL_ LAGHER's Lemma, as follows.
Lemma A.8.2. If f Is a continuously differentiable function,
8
Is as
above, and X+ 2 8 s xr s Y - 2 8, then 1
sR I f(xr)I2 s
-1. fXYIf(x)I2 dx + (fX If(x)I2 dx)
Y
'
(fX If'(x)I2 dx)l,
Theorem A.8.1, specialized to rational numbers a/q, immediately gives the following theorem. Theorem A.8.3.
2]
q5Q 1sa5q, gcd(a,q)=1
S(1)12 5 (nN + Q2)
a
q
I
12.
I
The factor (n N + Q2) may be replaced by (N + Q2).
Another kind of "Large Sieve-inequality" with weights is the following theorem.
Theorem A.8.4. Put
S
s r = min
s*r
isrsR
(N +
3, 2
S -11-1
r
J
III xr - X.
I
III .
Then
S(x ) 12 S
r
E Ia 12, M<nsM+N n
and
s
qsQ (N+
2
qQ)-1
lsa&q,gcd(a,q)=1 IS(a) q
2
Z
M<nsM+N
Ia I2. n
A.9. Dirichlet Series
331
A.9. DIRICHLET SERIES
A convenient reference for this section is
E. C. TITCHMARSH, The Theory of Functions, Oxford 1932. The well-known ABEL Theorem for power series: If an = s Is convergent, then f(x) = I an xn Is uniformly convergent In 0 s x s 1, and limX._1- f(x) = s, which may be formulated for complex x, too (uniform convergence Is then true In some angle), has a counterpart for the DIRICHLET series.
Continuity Theorem for DIUCHLET Series.
If the DIRICHLET series Z a n n-" is convergent In s = so , then It Is uniformly convergent In the SToLz angle I arg(s - so) I s 2 it - 8,
where 0
