Topics in arithmetical functions: asymptotic formulae for sums of reciprocals of arithmetical functions and related results

TOPICS IN ARITHMETICAL FUNCTIONS

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NORTH-HOLLAND MATHEMATICS STUDIES

43

Notasde Maternatica (72) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Topics in Arithmetical Functions Asymptotic formulae for sums of reciprocals of arithmetical functions and related results

J.-M. DE KONINCK University of Lava1 Quebec, Canada and

A. lVlc University of Belgrade Belgrade, Yugoslavia

1980

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

NEW YORK

OXFORD

North-Holland Publishing Company, 1980 All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright o wner.

ISBN: 0444860495

Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole distributors for the U.S.A.and Canada: ELSEVIER NORTH-HOLLAND, INC. 5 2 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

PRINTED IN THE NETHERLANDS

INT RO DU CT ION

The object of this monograph is to examine several topics in the

theory of arithmetical functions. The principal topic to be studied is that of asymptotic formulae for the sum

where f ( n ) is a non-negative arithmetical function, and the sununation is over those n not exceeding

P

for which f ( n ) * O

.

These sums possess

an intrinsic interest apart from their number-theoretic applications. Two of the most interesting classes of arithmetical functions, namely multiplicative and additive functions, require different techniques for the estimation of (1). Since the reciprocal of a multiplicative function is itself multiplicative, we see that for the case of multiplicative functions f , the estimation of (1) may be regarded as a special case of the estimation of

where g ( n ) is multiplicative. Since an extensive literature already exists for this problem, we will deal with reciprocals of multiplicative functions in Chapter 1 only. There, we use important analytic tools such

J.M. DE KONINCK AND A. IVIC

vi

as the convolution method and the method of complex integration in order to prove the basic Lemma 1.1, which is repeatedly used in later chapters. When f is additive, the sum (1) is much more difficult to estimate, and no significant results were hown for a long time. The first h o r n result is due to R.L. Duncan C11, who proved the inequality

(3)

A general method for estimating reciprocals of additive functions

was introduced by the first named author in his 1972 doctoral dissertation at Temple University, Pa., parts of which appeared in De Koninck C11. This method, which rests on the fact that zf(n)

is multiplicative whenever f

is additive, is explained in Chapter 2 and yields improvements of (3) to

(4)

Z I

n a

l/w(n) = A1x/loglog3c t.. .tA~/(loglogz)'t

O(x/(loglogz)'tl)

,

where IV is arbitrary but fixed, A 1 = l , and the remaining Ails are computable. Subsequent papers, a number of which are due to one or both of the authors, contained further results which give more than enough material for a systematic account of sums of reciprocals of arithmetical functions. This topic seems particularly well suited for a monograph such as this, since it allows treatment in considerable depth without being too wide in scope. Furthermore, this branch of analytic number theory is quite new, and no general self-contained publication has yet appeared on this subject. We have included a number of our hitherto unpublished results, as well as some sharpened asymptotic formulae. In particular, we improve (4)

vii

INTRODUCTION

in Chapter 5 to

where M

is arbitrary but fixed, L .(z) ( j = 1,.. .,M) is a slowly oscil3

lating function admitting an asymptotic expansion of the form

( 6 ) L j (z) a

.(log log z)- '+. . .t aN,3.(log log x)4

1J3

t O(

(log logxrN-l)

,

where all the constants are computable, and N is an arbitrary but fixed integer. Although no claim is made that (5) is the best possible asymptotic formula, it certainly will be difficult to improve. A large number of additive functions arise from the logarithms of

positive multiplicative functions. In Chapter 3 we give asymptotic formulae for reciprocals of logarithms of some of the most important multiplicative functions. In addition we establish asymptotic formulae for reciprocals of many interesting additive functions. The methods we develop are also used t o estimate

(7)

where both g and f are additive; this is the subject considered in Chapter 4. We have devoted Chapter 7 to the study of reciprocals in "short" intervals. These are sums of the type

J . M . DE KONINCK

viii

AND

A. IVIC

where the interval is "short" in the sense that h

o(z)

as x+-

.

The

method used allows us to obtain estimates for other sums as well, such as

(9)

A (z,h) =

4

1 1 , zwath,f(n) =q

where q is a fixed integer, and f is a suitable arithmetical function. This estimate is established by using both general results and methods of analytic number theory. We hope that digressions of this sort will make the book more interesting for the general reader. Even in these digressions, "reciprocals of arithmetical functions" remains the thread which

holds the whole together. Analytic methods are also employed in Chapter 8 to estimate

where f belongs to a certain class of non-negative additive functions, and

9

is a suitable subset of the set of natural numbers. It is only

in Chapter 6 that the analytic approach is abandoned, and special elementary methods are used to deal with large additive functions such as

, B(n)

1

ap

.

Apart from the asymptotic formula (Theo-

Pal In

rem 6 . 2 )

the results are not as sharp as those obtained in other chapters, although we are certain that sharper results (such as those stated as open problems in Chapter 9) can be obtained.

INTRODUCTION

ix

This monograph does not resolve all the major problems connected with asymptotic formulae for reciprocals of arithmetical functions. In fact we give a list of open problems in the last chapter. It is our modest hope that this monograph will induce further research in this interesting field. Although this book is intended primarily for specialists, we think that it will be of interest to the more general reader as well. Apart from a general knowledge of analytic number theory and the theory of arithmetical functions, which may be found in a number of standard texts such as Hardy and Wright C11, Ayoub 111, Grosswald C11, etc., the reader needs only to possess a basic knowledge of calculus and of complex analysis. The text is, with some minor exceptions, completely self-contained; we have tried to make the exposition as clear as possible, without omitting important details, by deleting routine calculations and repetition of similar arguments. Each chapter is followed by a section of Notes, where all necessa-

ry clarifications and discussions are given, together with appropriate references. We have tried to include in these references all the papers which deal with asymptotic formulae for reciprocals of arithmetical functions and related topics. The notation is kept standard throughout and is fully explained in the next section. A number of mathematicians have kindly read the manuscript and have made many useful critical remarks and suggestions. We take this opportunity to thank Dr. E. Brinitzer-Scriba, Professor P. Erdos, Professor

J.M. DE KONINCK AND A. IVIC

X

E. Grosswald, Professor G. Lord, and Professor H. Lord.

We would like to express our appreciation to Facult6 des Sciences et de Ggnie, Universitd Laval, Quebec, and to Mathematical Institute of Belgrade and Repub. Zaj. of Serbia for their financial support of the technical preparation of the monograph. Finally we wish to thank Ms.Louise Papillon who typed the final camera-ready text with considerable care and speed.

Jean-Marie De Koninck

Aleksandar Ivi6

Mpartement de Mathhatiques

Rudarsko-geolozki fakultet

Universitd Laval

Universiteta u Beogradu

Qugbec, G1K 7P4

Djugina 7, 11000 Belgrade

Canada

Yugoslavia

NOTAT I ON

Owing to the nature of this monograph no attempt has been made to secure absolute consistency in the use of notation. The following summary explains some of the most common symbols and functions used; all other necessary notation will be specified within the body of the text.

natural numbers (positive integers).

k , Z,m,n:

p:

a prime number (without exception). the greatest integer not exceeding the real number

[z]:

1

nS?:

n

the empty sum is defined to be equal to zero. : a product taken over all primes p

not exceeding z ; the empty

product is defined to be equal to unity.

1' g ( n ) / f ( n ) :

na

z

n:

a sum taken over all natural numbers n not exceeding

for which f ( n ) z 0

.

a product taken over all primes.

P

(m,n) : the greatest common divisor of m dl n : d divides n

pal In : pa

:

and n

.

.

divides n

n:Z(modk) : k l ( n - Z )

f

.

: a sum taken over all natural numbers n not exceeding z ;

P a

d n

z

, but

pat'

does not.

.

a sum taken over all divisors of n

xi

(including 1 and n ) .

xii

J.M.

f

:

DE KONINCK AND A. I V I C

a sum taken over a l l primes that divide n

P n

d(n)

f 1 : the number of divisors of n

d n

.

.

dk(n) : the number of ways n can be written a s a product of

k

factors.

+(n) =

1 1 : Euler's t o t i e n t function, which represents the msn, (m, n ) =I number of positive integers coprime with n and less than n

.

a ( n ) : the number of f i n i t e non-isomorphic abelian groups with n elements. :

the sum of a l l divisors of

n

a(n)

i

n

: the number of d i f f e r e n t prime factors of

Pn G(n)

.

1

a :

I i:"" .

the number of t o t a l p r i m factors of n

Pal In

p ( n ) : the IGbius function defined by p(n) =

.

1

n=l

(-l)r

n=p,. . .p,,pi's different primes

otherwise

0

A(n) : the von Mangoldt function defined by

a

A(n) =

n=p otherwise

P(n) : the number of unrestricted p a r t i t i o n s of a positive integer n .

n

a(n)

p : the greatest square-free divisor of

n

P In 8(n)

T

: the sum of d i s t i n c t prime divisors of

P n

B(n)

1

up : the sum of a l l prime factors of

n

n

.

.

.

Pal In B1(n) =

1

p a : the sum of d i s t i n c t prime powers t h a t exactly d i -

Pal In

vide n

.

NOTATION

xiii

.

p ( n ) : the greatest prime divisor of n

e(x)

=

$(XI

=

~ ( x )

cm P c A(n) n*

PQ

11

PQ

:

*

the number of primes not exceeding x

.

x : a real variable. : complex variables (Res and

z,s

parts of

s

t = Ims)

.

~ ( s ):

Ims

denote the real and imaginary

respectively; common notation u

Res

and m

Riemann's zeta function defined for Re s > 1 by

~ ( s =)

1 n-' ,

n=l

otherwise by analytic continuation. x ( n ) : character modulo a fixed natural number k

x,(n) : principal character modk ; x,(n)

=

.

1 if (n,k)

1

otherwise.

zero

m

r ( z ) : the Gamma function defined for Rez > 0

by r ( z )

=

J

tz-' e-tat,

0

otherwise by analytic continuation. X

.

expx=e y

:

p

m

Euler's constant, defined as +

y

$(x,y)

c (log(1

P

- Up)

1

t

-

y

J

e logx.dx=O.5772157

-x

.

Up)

1 : the number of positive integers n not ex-

n5xc,p (n)59

ceeding x all of whose prime factors do not exceed y

f(x)

- g(x)

as x

f(x) = O(g(x)) constant

... .

+

xo means l i m f(x)lg(x)

=

.

1

33x

means D O

.

lf(x)

I

5

Cg(x)

Here f(x)

for x a o and some absolute

is a complex function of a real

xiv

J.M. DE KONINCK AND A. I V I e i s a positive function f o r z z o

variable, and g(z)

f(z)


1/2 , we obtain by partial

summation

for every A r O and every

E >

0 , which gives (1.31) after collecting terms

. .

x logl/k-i2 for i = 1 ,2,.. ,N

For another application of the convolution method we now consider a(n)

, the number of non-isomorphic abelian groups with

n elements. This

function is multiplicative and prime-independent, which means that a (pa)

, and not on p tricted) partitions of n , then

depends only on a

a@")

(1.32)

and in particular a ( p ) = 1 Dirichlet series of l/a(n)

t

p-s

P

-2s + ++ t

l

t ____

t

t

a (PIP"

-4s

-3s

t

,

= 2 for every prime p

n (1

n=l n(1 P

is the number of (unres-

.

Thus the

is

1 l/a(n) . n-S

A(s) =

If P ( n )

P(a)

, a(p2)

m

(1.33)

.

t...)

1 / 2

.

Changing

>

we see that (1.34)

A ( s ) = s(s)cs(zs)l-1~2w(s) ,

where

is absolutely convergent for Re s

>

1/3

.

From

we have by the uniqueness theorem for Dirichlet series

whence

s

to 2s

RECIPFWALS OF MULTIPLICATIVE FWCI'IONS

15

so that m

where

1 h ( n ) n-'

n=l

.

= [ 0 / 2

z(xU - 1)

1 > u logx > u2

t o x1 = u - l q 2 > 3/2 = (logu

rl

.

,

.

-1 -1

)

, and

Then

since f o r

we obtain

,

which gives log(1 3/2

t

x-l(xU - l ) - l ) log-hxax

l o g ( l / u ) - ~ l o g - ~ z 1

, log(1

t

u

-1

z (z

-

-1 u 1)-l) < z (z

-

1)-l




zulogz

t z

2

ulogz2 =

v ) 2 1 - v / 2 (0 < v < 1)

-1

.

with v

we deduce t h a t log(1

f o r z2

.

5 z 5 z

3

+

( . ( X U

and u

- l))-l)

- 1 ) ) - l (1

t ( . ( X U

small enough, and t h a t

u l o g r s ulogz3 =

U(0q

-2

)

o(n)

so that xu

-

- 1 < (1 t

rl)

u logz

.

,

n/2)

(xutl - 2 )

-1

,

J . M . DE KONINCK AND A. IVI6

20

X

3

x-l(logx)-h-lh

X

h-1(log-hx2-log-hx3)

= k - l u h - h -1 IT k t l

2

This gives

which combined with (1.45) proves (1.44). From (1.44) and (1.42) we obtain, as u + O t G(u)

,

u-l(log u -1) -1 t O(o-'(logu -1) - 2 log logo-')

which implies (1.40)

.

As already remarked, Theorem 1.4 follows then direc-

t l y from Lema 1 . 2 and (1.40).

RECIPROCALS OF MULTIPLICATIVE FUNCTIONS

21

NOTES

As remarked in the Introduction, the aim of our monograph is to study asymptotic formulae for

1'

(1.47)

n-

llf(n)

,

where f ( n ) is an arithmetical function of a certain interest. Therefore we restrain from studying

1

n*

f(n)

for multiplicative functions f

,

though this problem is one of the most interesting and important topics in analytic number theory, and we give estimates for (1.47) in the case of several interesting multiplicative functions f only. For the sake of completeness, however, we mention here the following sharpest known asymptotic formulae for some familiar multiplicative functions: (1.48)

(1.50)

The first two of the above formulae are due to A. Walfisz C11, while (1.50) is proven in the forthcoming paper of G. Kolesnik C11 (see Chapter 7 for a more extensive discussion concerning (1.50)).

As a general result concerning summatory functions of multiplicative functions we mention the following result of E. Wirsing C11, which

has wide applications and has initiated subsequent research: Let f ( n ) be a non-negative multiplicative function for which

22

J.M. DE KONINCK AND A. IVI6

for some constants c1

>

0

as x+m for some T T O

.

a 2

2

,0

5

c2 < 2

and all primes p

and integers

, and let

Then, as x+-

,

Formula (1.1) may be found in Montgomery C11. The more general sum

2'

nQ, (n,I ) =1

1/$(n)

is investigated in Halberstam-Richert C 1 1 , Chapter 3 ,

95, and has an application to the so-called Titchmarsh divisor problem of

estimating

I:

l 1 is a fixed real number.

CHAPTER 2 RECIPROCALS OF "SMALL" ADDITIVE FUNCTIONS

11. Introduction

An arithmetical function f is additive if ffmn) = ffm) + f f n J whenever m and n are coprime integers. The problem of finding an estimate for

1

n-

f(n)

, where f is additive, can be approached in the following manner:

29

J.M. DE KONINCK AND A. IVIC

30

For many well-known additive functions f the sum mated, the double sum order than x ( 1 ,

t

c;)

l2

.

This approach allows

1 f(n)

n<x

1, - x

converges, while

1, 1,

can be easily estit

1,

is of lower

to obtain fairly accurate estimates for

LIS

for a large class of additive functions f

, and

so, for example,

we obtain

1 w (n)

2

n 2

log log x

t p

x

0(x/log x)

t

,

(2.3)

where

1 (log(1

y t

P

P

- l/p)

-k

Up)

.

On the other hand, if f is additive, there is no obvious way t o obtain an estimate for sums of the type

1'

nSx

.

l/f(n)

The purpose of this

chapter is to show how one can obtain precise estimates for sums of reciprocals of "small" additive functions (among which we find the functions w

and Q ) . This class of "small" additive functions will be defined more

precisely in 53. Actually our approach will yield an asymptotic expansion of the form

1'

nSx

l/f(n)

x

a

1

Ai/(loglogr)i

t

i=1

O(x/(loglogx)atl)

,

where a is any preassigned positive integer and the Ails are computable constants depending only on f

,

We will also show that our method allows

LIS

to estimate certain

1' g(n) / f ( n ) , where g is multiplicative and f is nsc additive.. Finally we will extend our method to study asymptotic expansions

sums of the form

SMALL ADDITIVE FUNCTIONS for

52.

1'

nsx

l/(f(n))k

31

, for an arbitrary positive integer

k

.

The method.

If f is an arbitrary additive arithmetical function, then, for an arbitrary complex number t * 0 , t f ( n ) is a multiplicative arithmetical function.

If g ( n ) is a multiplicative function, it is, in many cases,

possible to estimate

1g ( n )

using complex integration.

1111:

The idea that we will use is the following: given an arbitrary arith-

Ift f ( n ) for real t~ C0,ll (where nsx the prime in the sum indicates that the sum is taken over all nlz: for

metical function f ( n )

which f ( n ) s 0)

.

, let

Suppose that

(2.4)

F(2,t)

with R ( x , t )

F(z,t) =

O(H(z,t))

G(s,t) t R(z,t)

holds uniformly for t e [0,11

and H(z,t) are integrable functions of t z+-

.

, and

, where

C(z,t)

~(z,t) ~(C(z,t))

as

Then, using ( 2 . 4 ) , we have

One would like to integrate the sum in ( 2 . 5 ) with respect t o t between 0 and 1 and get

J . M . DE KONINCK

32

AND

A. IVI6

Integrating i n a similar way the r i g h t side of ( 2 . 5 ) , one would hope t o get the desired estimate f o r

1'

l/f(n)

n a

.

In most cases, however, the r i g h t side of (2.5) w i l l be unbounded i n the neighborhood of between 0 and 1

t =0

, so

t h a t it i s impossible t o integrate it

. between

To overcome t h i s d i f f i c u l t y , we shall integrate ~ ( z ) and

1

, where

~(z) 0 -f

, as

z-fm

.

We then obtain

If we integrate i n the same manner the r i g h t side of ( 2 . 5 ) and show, by a proper choice of

~ ( z ), that f o r

z

s u f f i c i e n t l y large the second sum i n

( 2 . 6 ) is small compared t o the integrated r i g h t side of (2.5),

obtain the desired estimate of

1'

n a

l/f(n)

we then

.

An estimate l i k e ( 2 . 4 ) is not always possible f o r

F(z,t)

.

But

using a r e s u l t of A. Selberg (Lemma 2.1), we w i l l find a large c l a s s of functions f o r which G ( z , t ) on C0,ll

D ( t ) z logt-'z

, where

D(t)

is continuous

and H(z,t) = ~ l o g ' - ~ z, and t o t h i s class of functions we w i l l

be able t o apply the above method.

13. Selberg's r e s u l t and basic definitions.

Before we define the class o f functions t o which we w i l l apply the method outlined i n Section 2 , we w i l l prove Selberg's r e s u l t .

With our

SMALL ADDITIVE FUNCl'IONS

33

notation, and the r e s t r i c t i o n t o the p a r t i c u l a r case needed here, it may be stated a s follows: m

Lemna 2.1.

Let g ( s , t ) =

1bt(n)/ns

n=l

f o r Res

m

1 Ibt(n) 1n-l log3 2n

be uniformly bounded f o r

n=l

m

(s(s))tg(s,t)

=

uniformly f o r

It

Proof.

1

n=l

I

for

a,(n)/n"

s 1

u>1

.

It

I

5

1

.

u>1

, and

let

Next, set

Then

.

,x2 2

Using Lemma 1.1 with N = 1

, and defining

D,(x)

1dt(n) n a

we obtain

W e have, since

Ibt(n) I 17 (log 2n) 3

n=l


0

and E

>

0

, both

f o r a l l x > X and a l l

i t x)

t E C0,ll

independent of t e

=

C0,ll

o(

.

dt

.

.

Recall t h a t

R(r,t)

This means t h a t there e x i s t

t

, such

t h a t IR(x, t )I < B x logt-2x,

Hence, f o r x s u f f i c i e n t l y large,

X

(log log x)

1'

Combining (2.13), (2.18) and (2.19), we obtain

1' ?la

1

fo

which is the desired result.

Q

1

Ai

i=1 ( l o g l o g x )

+.

X

1,

ol(loglogz)Q+~

42

J.M. DE KONINCK AND A. IVId

15. Applications'of the main theorem.

With the help of Lemma 2.1, we will now apply Theorem 2.4 to some well known functions. Theorem 2.5.

1'

n*

where al

1

a

a

i c iJxT i=l (log logz){

1

, a2

= 1- p

X

(log log x)atl),

, and the remaining

3

ails are computable cons-

tants. Proof. It is clear that, for u > 1 and

It I

5

1

,

Let (2.20) 1 seen to be a holomorphic function of s for a > 2 Indeed, if a > l , we have

then

g(s, t ) i s

(where the argument es (2.21).

logg(s,t)

of logg(s, t ) is chosen so that

= t c log

P

- TI


0

Therefore g(s, t ) = elogg(s-'t) represents a holomorphic function of 1 for u > 7 (see Apostol C11 p. 394). As

s

satisfies the conditions of

mentioned by Selberg C11, g(s,t)

t w ( n )and we obtain

L e m 2.1; hence a t ( n )

(2.22)

.

nl?:

uniformly for It1

s

1

.

We can now make use of (2.22) in order to apply Theorem 2.4. We first have to verify that D ( t ) all a

.

It is easily seen that g(1,t)

tion (2.21). Finally, w(1)

Furthermore, l/r(t) 0

, and

w(n)

belongs to CatlCO,ll , f o r

g(l,t)/r(t)

t

Applying Theorem 2.4, we obtain

E

E

CmCO,ll

C"C0,lI

1 for n

f

, from the representa-

, and hence g(l,t)/r(t)~c~rO,lI. 1 . Thus w E Sa , for all a .

J . M . DE KONINCK AND A. IVId

44

a

a

c

i i=1 (log 1 o g x ) i +-

1 1' nsx Jx=

Ol

X

(log logx)atl

with

For al and a2

, we

(2.23)

1

al

have

D(1)

P

D(t)

D(t) m

-tD'(t) t*

t=l

From (2.21) and (2.22),

n

g w = g(1,l)

ew ( t )

m

t

p v=2 vp

t

I

1 1 (1 - -) (1 t -)

P

P-1

-

= D(1)

D'(1)

t=l

, with vtl v

(-1)

t

p

Hence

with

ri(l)

= -y

(for a proof of t h i s , see Landau Cll p . 138) and

= 1

1-D'(1)

.

SMALL ADDITIVE FUNCTIONS

w'(1) = -

1 1 (_,)V+l cp mc ,+cm+c c 7 up p p (p-1) v=2

v=2

1 1 P p(P-1)

1 Ilog(1 - -)p1

t

-11 p

1 { l o g ( l - -) P P

t

-11 P

D'(1) = y

{ l o g ( l - -) 1 P

P

=

45

t

~

-

;r n 1

.

Therefore t

P

t

-11 P

p

and (2.24)

a2

= 1 - p

.

W e w i l l now give a second application of Theorem 2.4.

denotes the t o t a l number of prime f a c t o r s of

n

, then,

If

n(n)

proceeding as i n

the proof of Theorem 2.5, we obtain Theorem 2.6. a

1

.(

X

(log log x ) a + l

where b, = 1

, b2

= 1 - 0-

1 e7 , and the remaining p(P-1

b i t s a r e com-

putable constants. From the preceding two theorems, we have the following corollary. Corollary 2.7. 1

n s iJ3i-J--

L

1

cx -@T= (loglogx)2

3

+ o~(loglogx)~

J.M. LE KONINCK AND A. IVIC

46

with

C =

1-P k1- 1 .

Proof. The proof is immediate from Theorem 2.5 and Theorem 2.6.

86.

A

generalization of the main theorem. Definition 2.4.

Let

g(n)

tf(n) = a,(n)

be the set of all ordered pairs of arith-

S;

which satisfy the following four conditions:

metical functions (g,f)

4)

,

satisfies the conditions of Lemma 2.1, with

~ ( t= )g(l,t)/r(t)

E

.

c~+~co,~I

From this definition, we observe that if (g,f)

uniformly for It I s 1

.

E

S;

, then

Therefore to each ordered pair (g,f)

E S;

we

can associate the function D ( t ) , and, using this function, we define the functions B i ( t )

and A i ( t ) , 1 s i s CL t 2 , as in Definition 2.3.

now state the following theorem. Theorem 2.8.

Let @,f)

E

S;

.

Then

We can

47

SMALL ADDITIVE FLINrnIONS

Proof: Since (g,f)

with R ( z , t )

E

Si

, we have

U(zlogt-2z) uniformly for t

E

C0,ll

, D(t)

E

Cat1CO,11

and

Now

We observe that

-.(

2

)

= o (

3:

(log log z)

(log-)2

lJ

and the proof of the rest of the theorem is entirely similar to the one of Theorem 2.4 and will therefore be omitted.

As an application of Theorem

2.8,

we now state Theorem 2.9, without

proof since its proof follows essentially the same lines as that of Theorem 2.5.

J.M. DE KONINCK AND A. IVId

48

Theorem 2.9.

where ~ ( n )is the Gbius function, d,

= 6/n2

6

, d,

= -

11 - p

n2

t

1

P

and the remaining d i t s are computable constants.

17.

Estimates for

1 l/(f(n))k

for an arbitrary positive integer k

nsx

In this section, we obtain estimates for

, and

f E S a

k

2' l/(f(n))k

n-

is a fixed positive integer, k s a

.

.

, where

We first make a defi-

nition and prove two lemmas that will be used in the next theorem. Definition 2.5. For t = Bi(t)

'Bi(t)

1,2,

for i

(0,11 , set 1A i ( t ) =

E

...,a t 2

; with A i ( t )

~ ~ (and t )

and B i ( t )

as in De-

finition 2 . 3 . Next, define 2

i-1

1

Bi(t)

(i-j-1)

j=1

and 2 s k

2

~ ~ =( (-l)i-2 t ) 2Bi(t) 5

a

and k A i ( t )

for i = 2 , 3 , ..., a t 2

.

More generally, for

, set

= (-l)i-k* B i ( t )

for i = k , k + l ,...,a t 2

sionally write k ~ i for k~i(l)

.

.

We shall occa-

49

SMALL ADDITIVE FUNCTIONS

Lemma 2.10. k s

CL

.

.

(0,ll

and

5

E

(OylI and Proof:

, depending

7

tz

be a positive integer, kAi(t)

5

p t 2

, such

only on f

that

uniformly f o r

k5iscit2 ;

2 ) there e x i s t s a constant N

t

k

Then

I kA i ( t ) I E

let

the corresponding functions

1) there e x i s t s a constant M

t

, and

f E S a

, associate

To f

k s i s a t 2

Let

ksisa

, depending

only on f , such t h a t

.

This lemma i s a generalization of Lemma 2 . 2 , and it follows

hnnediately from Definition 2.5. The following lemma is a generalization of Lemma 2.3.

Lemma 2.11. positive integer.

Proof:

Let

2 2

3

, and

let

~ ( 3 : )5 u

E(Z)

and

Let

u

1

.

Let

B

be a

Then

h(t) =

t t B

.

From the proof of Lemma 2 . 3 , it is

e a s i l y seen that the only two possible maxima of

at

5

, Therefore

h(t)

in

[ED)

ul

are

J.M. DE KONINCK AND A. IVIC

50

W e now e s t a b l i s h a general formula which w i l l help

Let f

Theorem 2.12. t r a r y positive integer,

I'

(2.25)

uniformly f o r u Proof:

E

f (n)

c

, 11

1

t

.

Let k be an arbi-

E

(x))

1

(logx)-20

The proof is by induction on k

and R1(x)

3

0(x(log log x)

, where ~ ( x =)

~ ( x 2) v s 1

(2.10) holds and we have, f o r

R(x,t)

x

P k + l(log logx)cL+l

CE(Z)

2

(1oglogx)i

i=k

3: logU-

let x

Then

=

(f(n))

n*

1

U

.

ksa

, and

Sa

E

us find an es-

.

Since f

E

Sa

, equation

,

a r e the functions defined i n the proof of Theorem 2.4.

R ( x , t ) = O(xlogt-2x)

uniformly f o r

Since 0 < ~ ( x 5) v

1

dt t

5

1

E

[0,11

, and

~ ( x ,)

R1(x)

,

logv - logE(x) = o(l0gE

E (XI

From the definition of

t

-1

(x))

.

0(-1g :2

*

SMALL ADDITIVE FUNCTIONS

51

Hence

v

I

(2.27)

O(x logt-2

T h a t R(x, t )

e x i s t a constant B IR(x,t)

I




0

Bx logt-2x

x) uniformly for t

and an X > 0

, for

, both

.

O(ZE(2))

E

C0,ll

means that there

independent of

a l l x > X and t

E

C0,ll

.

Therefore, i f

is s u f f i c i e n t l y large,


0

i s a r b i t r a r y , but fixed.

al and a2

i s given i n the proof of

Theorem 2.5, and similar computations of corresponding constants i n Theo-

rem 2.6 and 2.9 a r e therefore omitted. In Theorem 2 . 9 we could have written

For t h i s , and other elementary properties of the Mobius function

u ( n ) see Grosswald C11, o r Hardy and Wright C11.

CHAPTER 3 RECIPROCALS OF LOGARITHMS OF MULTIPLICATIVE FUNCTIONS

51. Functions with main term asymptotic to Cx/logx

.

In this chapter we shall be concerned with asymptotic formulae for nQ

1'

nsx

l/logf(n)

, where f is a positive, multiplicative function, and

denotes summation over those n not exceeding x for which f(n)

>

1.

Most well-how multiplicative functions will be seen to satisfy one of the following:

b)

(3.2)

n a

l/log f( n )

- Cf x/log logz

,

or

f is a positive constant depending on f .

where C

In this section we shall give an asymptotic formula of the type (3.1) f o r a large class of arithmetical functions which includes the functions g ( n ) and o ( n )

.

We begin by making the following

Definition 3.1. Let

be the class of multiplicative functions

f with the property 65

66

DE KONINCK AND A. IVI6

J.M.

, where

f o r all positive integers n

tisfying 0

1 and y

8 s

c

> a

.

, 8 ,y

For f

nf = rninh: f(m)

(3.5)

Q

let

F

E

are positive numbers sa-

1 for all m

>

2

.

nl

Because of multiplicativity, it is enough to verify (3.4) for n = p k , in order to establish that a given multiplicative function f

belongs to

.

$

number such that

$ , let

a=8

, and recall that

k

To see that 4 y >

u = 1

E

= 1 , let

4(p )

k

p (1

-

y

be any

Up)

.

Then

(3.4) reduces to

which is obvious since One

o,(n) =

and

y

i

dn

(1

-

l/p)Y s 1

y >

1

.

- l/p s

(1

-

l/p)-y

,

can also show that the generalized sum of divisors function

, belongs to 3

dX

any number such that

for X > 0

y >

max(1,X)

.

.

To see this let a = 8 X

If n = p k then

and (3.4) therefore reduces to (1

-

l/pX)Y s 1

.t

upX

t

...

.t

u pk X s (1

-

l/pX)-y

.

The inequality on the left is obvious, and the right-hand side follows from 1 because Y

>

1

.

t

l/pX t

. .. t u p k X

1/(1

-

UPh)

5

1/(1

- l/P3 -y

L0C;ARITHMs OF MULTIPLICATIVE FUNCTIONS

67

Now w e formulate Theorem 3.1.

1'

13.6)

n-

, then

If

f

E

l/logf(n)

=

5

.

(-1) m-1 F ( m - 1 )

1

F ( t ) = (at t 1)-l

Proof.

Let f

E

$

, and

for -l/y

nP (1 - l / p ) ( 1 .

(O)

(a log zlm

m=l

where the 0-constant depends on M

(3.7)

f o r every positive integer M

For - l / y

t

,

o(5/logMt15)

It I0

m

t

5

1 p - m ~ ( p m ) -am p )t) .

m=l

t

define

I0

(3.8)

If

E

> 0

, then

f o r some positive constants

el = el(€)

and

c2 = c ~ ( E , B ) we have

(3.9)

l/pB

t

l/(pB - 1)

I c 2 p-'+'I2

f o r p s el

,

f o r p > c1

.

and (3.10)

l/pB

t

l / b B-

1)

Now f o r every integer m

(1 - l/pB)Ylt

I

5

2

I p-BtE'2

0p(prn)p -am)Itl

so that we obtain from ( 3 . 8 ) , f o r m ht (P")

(3.4) yields

0

(f (pm, p - 9 -

2

It'

1

I

(1 - l / p B) - Y l t l

, - (f(pm-l) P - a ( m - l ) ) - It I

,

68

J.M. DE KONINCK AND A. IVIE

ax

where we have used that f(x) that

yltl s

-

-x is non-decreasing for a

a

>

1

,

, (3.9), and (3.10).

A lower bound for ht(pm) may be proven similarly; the multiplica-

tivity of ht

where a ( n ) =

now gives, for some

np.

,

c3 = c 3 ( ~ , $ )

From Theorem 1.4 it follows that for every 6

P In

and therefore for 6 = d 2 we obtain

Partial summation gives m

By using the kbius inversion formula, we obtain from (3.8)

> 0

L 0 G A R J . m OF MULTIPLICATIVE FLJN(TI0NS

69

Using (3.12) and (3.13) we have

Suppose that -l/y

5

t

5

.

~ ~ m a x ( O , B t a l y - l ) Then we obtain uniformly i n

t

0

m

Since ht

is multiplicative and

1 ht(n)/n

n=l

i s absolutely convergent, t h i s

may be written a s

I ,

where F ( t )

is given by (3.7).

W e w i l l now deduce (3.6) from (3.15).

By (3.5) we have

O(1)

and theref ore 0

0

,

J.M. DE KONINCK AND A. IVId

70

To estimate the last sun observe that, for n

2

no

, f(n)>n

a/2

by (3.4), so that (3.15) yields

Using (3.15) we obtain

1' l/logf(n)

(3.16)

7

= x

nsx

F(t)xatdt

+

O(xl-BtE

.

t x

-l/Y

Since F ( t )

1

integration of

, partial

is infinitely differentiable on C-l/y,Ol F(t)zatdt

in (3.16) gives (3.6), for

E

>

0 suffi-

-l/y

ciently small. We now present another class of multiplicative functions for which

1'

ne

behaves asymptotically as c x/log x

l/log f (n)

f

.

Our approach this

time will be of a more elementary nature, and the obtained asymptotic formula (Theorem 3.2) will not provide as sharp an estimate as Theorem 3.1. Definition 3.2. A multiplicative function f belongs to the class

&I

if for every prime p

numbers a l J k

,

a2,k

9

*

and every positive integer k there exist

*.$a

kJ k

such that

(3.17)

where -1

5 aiJk 5

K uniformly in i and k with some K > 0

.

LOGARI'IHMS OF MULTIPLICATIVE FtbiCI'IONS From t h i s d e f i n i t i o n it is obvious that

f(n)

71

is s t r i c t l y p o s i t i v e

s integers ( i f t h e ai,k's and t h a t f ( n ) i s an integer i f the u ~ , ~ ' are were allowed t o take integer values l e s s than -1 then f

would not

always be p o s i t i v e ) .

$(n) and ~ ( n ,) it i s easy t o check t h a t

In addition t o

8

contains other well-known multiplicative functions such as: t h e Dedekind's function

$(n)

n

(1

t

, the

l/p)

P In function

$*(n)

n(-l)w(d)/d 1 dln, (d,n/d)=l

sum of divisors function

u*(n)

r e l a t e d t o the functions

o(n)

a r e a l s o contained i n function f

unitary analogue of Euler's t o t i e n t

3:

1 d d In, (d, n / d ) =I

unitary analogue of the

, and some other functions

.

and a*(n) While a l l of these functions

6

of Definition 3.1,

a l s o contains the

defined by k k f(p ) = p - p k - 1

f(2k)

, since

which does not belong t o f o r n = 2k

, the

and every integer k

which i s c l e a r l y impossible f o r k

t

-...- p-1 1 and (3.4) would give then

1

+

, since

the left-hand s i d e is a

p o s i t i v e constant.

On the other hand 5

means t h a t neither Theorem 3.2.

%

If

does not contain

5

nor

f

E

b

, and

Q

o,(n)

for

x

s 1

, which

.

ak,k 2 - 1 / 2

for k r k o

, then

72

J.M. DE KONINCK AND A. IVId

(3.18)

1'

nQ

= 5

l/logf(n)

log x

log log log x

The proof of Theorem 3 . 2 is based on the following elementary

8,

Lemma 3.1. If f E and

C2

then there exist positive constants

and a natural number nl

(3.19)

f(n)

C1

such that

< Cln(loglogn)K

for n

t

nl

,

and (3.20)

f(n)

2

C2rn/loglogm for rn

>

1

,n

= 2k r n , rn

odd,

where K is the constant appearing in Definition 3.2. Proof of Lemma 3.1.

If r = w(n) and p

T

since f is multiplicative

= n

and

(it-) K P-1 P In

,

is the r-th prime, then

LOGARITHMS OF MULTIPLZCATIVE FUNCTIONS W e now use the elementary estimates pn s n3’2 and log P

n

2

P In

p)

(valid f o r n t 3)

(which i s v a l i d f o r n

log ~ ( n )5 21og log n

follows from n

73

t

5

,

,

and which

t o obtain, f o r n t n 1 ’

s n e ~ p ( l o g ( ( 3 E )( l~o g l o g n ) K

which proves (3.19) f o r C1 = (3B)

( 3 ~ ) ~ (log n . 1ognIK

,

.

K

To prove ( 3 . 2 0 ) we note t h a t by (3.17)

f@k)

pk

- pk-l- ...-p- 1 ,

so t h a t f(p k ) = 1 can occur only f o r p

2 ; otherwise,

f(pk)

>

1

,

and we have

Since we have

log

log 1 / ( 1 -x)

1

x

+

x2 f o r

(1 - h 1 1 = f log(1 P Im P m

5

for m > l

5

and C4

1 - 1-P-1

log(C3 log logm)

s u f f i c i e n t l y large, so t h a t

0 s x -1

5

,Im[

+ o(1)

1/2

, we

1

p-l

obtain 1

+

02)

s log(c, log logm)

J.M. DE KONINCK AND A. I V I d

74 -1

where C2 = C4

which, when combined with (3.21)

)

proves (3.20).

1 l / p and p n would lead t o explicit psx C1 and C2 but C1 and C2 would still depend on K . Taking

Sharper estimates of values of =

Pi P2

nl '*

)

)

primes we find t h a t t h e

t o be the product of the f i r s t k

'Pk

bounds of (3.19) and (3.20) a r e actually attained, Proof of Theorem 3.2. there a r e O(1ogx)

1

the fact t h a t

numbers not exceeding x l / l o g n xflogx

292.2

1'

(3.22)

U l o g f(n)

2

%X

2

1 can occur only f o r n = 2k

Since f ( n )

lfl / l o g n t O

1'

nCx

t

O(x/log2 x)

, and

log2 x

)-

.

Using

(3.19) we obtain

1;x

log2 x

which gives the necessary lower-bound inequality. bound note t h a t i f

1

I/(log n t log c1 t log log log n)

xlogloglogs

n<x

f o r which f ( n )

k

k > k o and f(2 ) z 1

)

To obtain the upper

then f(Zk)

3/2 by

2

(3.17) and

.

5-1/2 If m 2 3 i s odd, then from k, k l/logm - l / ( l o g C2 t log m - log logm) = O(log log logrn/log2 m)

a

1'

n2

l / l o g f ( n ) O

and R e s > l ,

m

where H(s) = gives

1 h ( n ) n-' n=l

is absolutely convergent for Re s

>

1/2 . This

QUOTIWS OF ADDITIVE FUNCTIONS

109

which is obtained from Lemma 1.1, by observing the condition can be replaced by

) zI

5

A

IzI s 1

.

Note i n this connection t h a t t h e analogous estimate

c- 1 nsx holds f o r

0 2

.

The reason f o r this is t h a t the

corresponding Euler product f o r C > 2

does not converge absolutely f o r

1t

Re s > 1 only, since

t

c22-2s t

. ..

converges absolutely f o r R e s > l o g C / l o g 2 > 1 . In case C = 2 t h a t f o r some s u i t a b l e constants

1 2n(n)

nSx

one can show

D and E we have

DEC h g 2 x

t

Ex l o g x

t

O(x)

.

A proof of t h i s formula may be found i n the paper of E. Grosswald c21.

This Page Intentionally Left Blank

CHAPTER 5 A SHARPENING OF ASYMPTOTIC FORMULAE

11. Introduction The purpose of this chapter is to sharpen some of the asymptotic formulae proven in the previous chapters. In particular, we improve the formulae

and

by introducing new leading terms. Our results will hold for certain classes of non-negative, integer-valued additive functions f for which we shall give sharp estimates of the sum

1'

nsz

.

l/f(n)

These estimates will be seen

to depend on two deep lemnas which give estimates for the sum

1zf(n) nsx

,

and which possess an intrinsic number-theoretic significance of their own. Both lemmas (due to H. Delanee) were originally derived with the purpose of estimating sums of the form

1

1

n % , f ( n ) =k

, where

k (21)

is a fixed

integer, and f belongs to a certain class of additive, non-negative and integer-valued functions. As corollaries to lemmas 5.1 and 5 . 2 we shall deduce sharpest known asymptotic formulae for the sums 111

J . M . DE KONINCK AND A. IVI6

112

1

1

nsx, w ( a )=k

and

c

1 .

a%, n ( n )-w ( n )=k

Because of the unifying principles underlying these lemas, namely, the convolution method and complex integration (carried out in detail in the proof of Lemma l.l), and because of our desire to keep the exposition as clear as possible, we consider it more appropriate to devote a chapter to the sharpening of the asymptotic formulae, rather than to have stated the best results in the earlier chapters. Before proceeding further we shall introduce the concept of a slowly oscillating function. A real-valued function f(x)

is called

slowly oscillating (or slowly varying) if it is positive and continuous and for every c > 0 ,

for x 1. xo

(5.1) Many functions that appear in error terms in asymptotic formulae for arithmetical functions such as logA J: , exp(C 10gl’~x)

, log log x etc.

are easily seen to be slowly oscillating. These functions possess a canonical representation of the form

where

and 6 ( x ) are continuous for x z x o

p(x)

lim 6 ( x )

= 0

every

>

so

x+-

.E

0

.

lim

p(r)

x+-

that the above representation yields L ( x )

A > 0

1/2

f o r which

(5.2)

if t h i s set is non-empty (and u0 (p) =

of a l l

p 2

0

the set E

f o r which

(finite or

e x i s t functions A, ( z ) Ao(0)

(5.3)

A1(0)

n-

...

t-).

1


1 be t h e supremum of

Then f o r every fixed integer N (z

I

t

0 there

< 1 such t h a t

, and

= AN(0) = 0

N

(-1 A3. ( z ) l o g - j z

t

J =O

where the 0-constant is uniform f o r zf ( n )

, and

m

, A1 ( z ) ,. .. ,A N ( z ) analytic on

z f ( n ) = z(logz)z-l

Proof.

t

Iz

I

< 1

O(l0g

-N-1

x))

,

.

is multiplicative since f ( n ) is additive,

therefore follows that f o r Re s

>

1

and f o r

12

I

p L E

(since

It

uo ( p ) < 1)

114

J . M . DE KONINCK

AND A. IVIC

where

W e now show t h a t the i n f i n i t e product i n (5.5) i s an analytic funct i o n of

and z

s

for

IzI 5 p

E

E

and Res > u l > o

(p)

.

To investigate

the convergence properties of i n f i n i t e products we note the following s i m ple result:

Suppose u,(x)

and v n ( x ) a r e two sequences of complex func-

tions defined on the same s e t A

, and suppose that f o r every

n

2

1 and

for x e B _ c A

where

un

and

(5.7)

vn

a r e positive constants such t h a t m

m

n=l

n=l

Then the i n f i n i t e product

is absolutely and uniformly convergent f o r for x

E

B

.

To see t h i s , we define w n ( x )

by

3: E

B

and i t s value is bounded

A

If

V > 0

115

SHARPENING OF ASYMPTOTIC FORMJLAE

is such a number that

then there e x i s t s a M

>

0

and

Vn 5 U

such t h a t f o r

Iz

I

2

Vn

5

U

for n

2

1

u

Since

we see t h a t f o r n

where

W,

V

= Me V;

2

1 and f o r x

+

MVn

, so

E

B

that (5.7) implies absolute and uniform con-

vergence of the product (5.8) f o r x

E

B

.

This product is bounded by

m

Numerating t h e sequence of primes

p1,p2,. . . , p , , .

..

we note that

the above r e s u l t holds f o r

provided t h a t (5.6) and (5.7) hold with n replaced by p Recalling t h e d e f i n i t i o n of

where

G

, we

have

.

,

116

J.M. DE KONINCK AND A. IVId

(5.10)

If

IzI 5 p

E

and Res 2 o1 > u

E

then, since zf@) = z

(p)

,

we have

and

since ul

>

1/2 and

1 ( 1 pf @ k ) p - k u l ) z

p

converges by ( 5 . 2 ) for u1 > 1/2.

k=2

Moreover

which means that

1 VP

P


u 1

0

(p)

.

Therefore the product in (5.9) is absolutely and uniformly convergent in the region defined by

Iz

I

5 p E

E

and Re s > u1

the factors in (5.9) is an analytic function of

f is integer-valued and non-negative, tion of s

G(s,z)

s

.

Since each of

in this region, because

represents an analytic func-

in the same region. In particular, since R > 1 by hypothesis,

we find that for IzI s 1

where each g ( n , z )

is analytic on

IzI 5

1

.

117

A SHARPENING OF ASYMPTOTIC FOFMJIAE

From (5.4) we obtain

where

is t h e generalized d i v i s o r function defined by (1.10). Lemma

dz(m)

1.1 y i e l d s uniformly f o r

1zf(n)

(5.13)

n-

For From (5.5)

G(s,z) =

Iz

, we

I

5

= z

n

JzI

5

1g(n,z)

1

N

1 c i ( z ) logz-ix/n

n-'I

1 and f o r every n a t u r a l number

I

we have

k

(z) I

s1

.

have

n(1-zp-'t (i)pm2'

whence g ( p , z ) = 0 and

t

Ig(pk,z)I

implies that uniformly f o r

1 g(n,z)n-llogAn

nsx

Re z-N-1 O(1og

i=1

P

(5.14)

t

IzI

5

. . .) (1t zfS

k d(p ) = k

t

t

.fb2)p-2s

1 for

s 1 and every A

k

0 and

2

.

2

2 E

>

m

=

1g(n,z)n-llogAn n=l

t O(z

since

From (5.14) we obtain uniformly f o r

IzI s

1

logA z)

t

. . .)

This

0

,

,

118

J . M . DE KONINCK AND A. IVId

as follows: Writing

1


1/3 for which

(5.18)

if this set is non-empty (and u be the set of

p2

0 for which

1

otherwise).

+m

, and let

< 1/2

+m)

that F ( 0 ) = 6/n2 , A o ( 0 )

(5.19)

u,(p)

=

Furthermore let I R>1

. Then for every.fixed integer A . (2) ,... , A N ( z ) which are analytic on

mum of I (finite or functions F ( z )

(p)

=

...

AN(0) = 0

z F ( z ) +z1’2 logz-2z(

ncz

be the supre-

N > 0 there exist

I

(2

, and

N

1 A j ( z ) log-jr t O(log-N-lz))

j=O

where the 0-constant is uniform for

1.~1 s

Proof. We begin by noting that

1

.

u,(p)

R

and the A . ( z ) ’ s that 3

appear in both lemmas are not necessarily the same functions. For Res > 1

s 1 such

and

IzI

5 p E

I (since

o,(p)


1

.

In p a r t i c u l a r g,@) = 0 and

W e thus have

(s

- 1)

, we

~ ( s )

define H(s,z)

by

in order t o obtain (5.26)

G(s,z)

= H(s,z) ( s - 1/2)l-'

W e now give a brief sketch of the proof.

. I t consists of

121

J.M. LIE KONINCK AND A. IVIi

122

(a) proving that the function V ( s , z ) .s

and z

for

Iz

I

for < 1

12

I

5 p

, since

E

R>

I

and for Res > u l

is an analytic function of (and, in particular,

> u,(p)

1 by hypothesis),

(b) using the inversion formula ctiT

X

with c > 1/2

, and

(5.28)

to obtain

for some suitable functions $ z and Qz

, where Gz(x)

is not G(s,z)

of (5.26),

1zf(n)

(c) recovering

nlz:

and

from Gz(x)

(d) showing that the functions F,A,,

by a convolution argument,

., .,A N

satisfy the conditions

specified in the lemma. We begin with (a).

From the definition of h

P

and from (5.23),

we have that

for

s

P

.

m

Thus

1 g,(P

k=2

) p -ks

converges absolutely and uniformly

A

for I z I s

p E

SHARPENING OF ASYMPTOTIC F O W

I and for Res 2 ul > u,(p)

123

, since, by hypothesis,

m

converges. Therefore all factors of the infinite product

k=3

(5.31)

P

k=2

are well-defined for these values of s and

, and the general term may

z

be written as

where

Furthermore,

Since the sum in (5.18) is bounded we have

Thus

1 U2P


1 by hypothesis.

In applying the inversion formula (5.27) it should be noted that which appears i n (5.25) is analytic i n any open neighborhood of

Z(s)

, that

1 which is f r e e of zeros of

s

~ ( s )

Z(1)

= 0

and t h a t

.

exp(Z(s)) = (s - 1) ~ ( s ) To obtain (b) we replace the contour of integration

Cc

- iT , c + i T 1 by the contour used i n the proof of Lemma 1 . 2 ,

with the point

1 replaced by

s

s =1/2

.

(This is done because t h e

generating function now has a singularity a t s = 1 / 2 i n Lemma 1.1.)

From (5.25), from t h e product representation given i n (5.24)

, and

for

V(s,z)

for

IzI 5 1

,

and not a t s = 1 as

It1

from (1.14), it is e a s i l y seen t h a t

2

tl and u

2

1/2 - a / l o g ( I t l ) , a > O

.

The evaluation of the integral appearing on the right-hand side of (5.27) i s analogous t o the evaluation performed i n the proof of Lemma 1.1, and therefore we omit the d e t a i l s . Gz (t)d t

, the

To recover Gz(z)

same simple Tauberian arguments used i n Lemma 1.1 are again

used t o yield (5.29), where, f o r some c > 0 IzI s 1

(5.33)

from

, we

have uniformly f o r

A SHARPENING OF ASYMPTOTIC FORMULAE

125

and

(5.34)

t

For

Iz

I

s

1

1 2ni

)

stl

~(s,z)(s-1/2)~-~. .-. 5

ds

.

, @z(x) is an infinitely differentiable function of

x whose derivatives are obtained by differentiating under the integral

sign, and

is the circle

y,

q

Is

- 1/21

= P

minus the point s = 1/2 - r

,

is as defined i n Lemma 1.1.

We now prove (c) by using (5.22) and the convolution method. If y

y(x)

is a function satisfying 1

where F ( z )


1 / 3 +

i n which z 121 6

1

E

respectively.

I

1

, Re s 2

1/2

and

t E

We a r e thus faced with a convolution problem

i s not an integer, but an a r b i t r a r y complex number s a t i s f y i n g

, thereby

and s = 1 / 2

/z

and v ( s, z)

U(s,z)

producing an algebraic singularity a t the points

respectively.

s

1

This i s why a complicated contour of integra-

t i o n (introduced i n Lemma 1.1) w a s used. The above mentioned papers of H. Delange contain many results, re-

ferences and generalizations concerned with sums of the form where k

is fixed, and f is an additive function.

ginal paper concerning the problem of estimating R6nyi [I]; f o r the special case where k

1

,

For A. R6nyi's o r i -

c

~SZ,R (n)--w

, see

1 1 n 1

when one applies Lema 5.1 o r 5.2, and, though t h i s condition could doubtl e s s l y be somewhat relaxed, it is an easy one t o verify and furthermore it covers t h e i n t e r e s t i n g cases of "small" additive functions such as w ( n ) and n(n)

.

"Square-full" numbers are a special case of the more general 'powerful" numbers.

For a fixed k

E

N

one may define the set of powerful

numbers G(k) as G(k)

k { n c N I (pln) => (p I n ) )

Square-full numbers a r e then j u s t

G(2)

.

.

I f we s e t

and

then A k ( x )

i s the number of powerful numbers n

i n G(k) t h a t do not

146

J.M.

exceed x

. m

Fk(s)

where G

Since for Re s > l / k

1 f,(n) n-' n=l k

(6)

DE KONINCK

1 (1 + p - k s

AND

A. IVId

we have

tp-(ktl)s

P

t

... )

= c ( k s ) G,(s)

has the abscissa of convergence equal t o l / ( k t l )

, the

con-

volution method inunediately yields

and

Powerful numbers were first investigated by P. E r d k and G. Szekeres CIJs where the above formula w a s proven.

Much sharper asymptotic for-

mulae f o r A (x) can be obtained by more i n t r i c a t e methods; f o r these k methods see P.T. Bateman and E. Grosswald C11, and Ivi; C3l.

CHAPTER 6 RECIPROCALS OF "LARGE" ADDITIVE FUNCTIONS

11. Introduction In Chapter 2 we studied asymptotic formulae for the sum

1'

n-

where f ( n ) was a "small" additive function belonging to the class

l/f(n), Su

of

Definition 2.2. Broadly speaking, one could say that a general "small" non-negative arithmetical function f ( n ) is a function for which

as

z

-+

, where

L(z)

is a slowly oscillating function. This implies

that one can think of L ( z ) as the "average order" of f ( n ) in a certain sense. For every

E

>

0 we have

which means that the average order of f

is small. On the other hand,

there exist non-negative additive functions f for which (6.1) does not hold, and furthermore for every D

>

0 one can easily find a non-negative

additive function f satisfying

(6.3)

147

1 48

J . M . DE KONINCK AND A. I V l 6

which means t h a t f must be i n some way very large. t o be the average order of

More precisely, one can define g ( n ) f(n)

if

(6.4)

as z

-f

m

, where

t i a l etc.).

is a "well-behaved" function (polynomial, exponen-

Thus we can say that f ( n ) i s "large" i f there is an

such t h a t f o r n

where g ( n )

g(n)

2

E

> 0

no(€)

s a t i s f i e s (6.4).

For our purposes we w i l l formalize the concept of "large" additive functions by considering a class h'

, which

contains the most interesting

"large" additive functions t h a t we have i n mind. Definition 6.1. let (6.5)

(6.7)

and

For K

fixed,

y >

0 and

6

a fixed r e a l number,

149

LARGE ADDITIVE FUN(TI0N.S

The class H of large additive functions is defined to be the class of all possible functions f , F and F1 obtainable by varying K, y

>

0 and 6

.

Our definition of class H

includes for K

= y =

1,6

0 the

functions

(6.10)

and

(6.11)

The functions f3

, B and

B1

are of great intrinsic interest.

For example, for a fixed integer rn the number of solutions of B ( n )

=

m

is the number of partitions of rn into primes (not necessarily distinct), the number of solutions of f3(n) = rn , p 2 ( n )

=

1 is the number of parti-

tions of rn into distinct primes, while the number of solutions of Bl(n)

rn

is the number of partitions of m into powers of distinct pri-

mes. The function ~ ( n )is the additive analogue of the multiplicative function a ( n )

fl

p

, whose sum of reciprocals was investigated in

P In

Chapter 1. The average order of

~ ( n )and B(n)

(in the sense of (6.4))

is a2n/(610gn) ; this can be seen from the asymptotic formulae

150

J.M. DE KONINCK

AND A. IVId

(6.12)

and (6.13)

Our goal i s t o obtain estimates f o r

where f

,F

i f and only i f

and F1 belong t o H n

1

, and

(note t h a t

the same f o r F

f(n) 2 0

and f ( n ) = 0

and F1 ) . The techniques

based on the properties of the generating series

or

which successfully worked i n previous chapters, seem t o be of no use here. The d i f f i c u l t y l i e s i n the f a c t that there is no obvious way ( i f any a t a l l ) t o factor out a power of the zeta function from the generating s e r i e s .

W e therefore abandon t h e approach v i a Dirichlet s e r i e s and proceed instead with investigations of a more elementary nature.

W e shall not be able t o

obtain asymptotic formulae f o r sums of reciprocals of functions belonging

t o H , but only good lower and upper bounds f o r these sums. Estimates furnished by Theorem 6.1 extend, of course, t o

B

,B

and B1

, and

53 we give estimates f o r sums of quotients of these functions, and an

in

LARGE ADDITIVE FUNCITONS

151

asymptotic formula for

as well.

Bounds for sums of reciprocals

12.

Theorem 6.1. Let f , F and F 1 be functions belonging to class H

of Definition 6.1. Then there exist two positive constants 0

< el < c2

such that if

(6.14)

(6.15) then (6.16)

(6.17) and (6.18) where c1 , c2 and the 0

, we

u(n) O

then for x sufficiently large, and the lower bounds of Theorem 6.1 follow then from (6.28) and (6.29). To establish the upper bounds it is enough to establish the upper

bound in (6.16), since the other two proofs are identical. We now have for y y(x)

>

that

x

because

in the second sum, since, as previously noted, h ( x ) is increasing for x>x

.

For the function

$(X,Y)

1

1

n=, P (n)sy

we use the following estimate due to N.G. de Bruijn Cnl:

(6.31)

$(x,y)


L

This + l i e s

...+ ( a i - 1 ) p i t B ( n )

5

i s summed,

that

L(alt

...t ai - i ) t @ ( n )

Therefore we have

where we have used (6.38). From (6.47), (6.48), (6.49), (6.50), (6.51), and (6.52) we theref o r e have (6.53)

s

s

O(ktc/L)

3: t

+

O(x exp(-C3k))

.

.

O ( ~ log L 3: exp (-c4 (log x log l o g s ) '/*)I

Noting t h a t t r i v i a l l y

(6.54)

t

S 2

x

t

O(1)

, and

. log l o g r ) 1 / 2

k

(logx

L

exp(C,-(logx

choosing

Y

and (6.55)

. log log x)'I2

,

.

J.M. DE KONINCK AND A. IVId

160

where

C5 = C,/2

, we

obtain (6.43) from (6.53).

Theorem 6.2 can be e a s i l y generalized a s follows: Theorem 6.3.

If

t i v e constants C1 , C2

r > 0

is a fixed number, then t h e r e e x i s t posi-

such t h a t

(6.56) and (6.57)

Proof.

The proof of (6.56) is almost i d e n t i c a l t o t h e correspond-

ing proof of (6.43)

except t h a t i n S1 t h e sum i s over those

which B(n) < k l I P p(n)

and i n S2 t h e sum is over those

n

for

n f o r which

B(n) 2 klIr B(n) . This leads t o an estimate similar t o t h a t i n (6.53),

where

k

i s replaced by k1Ir

. The choice

k = (logx

. log l o g z ) r/2

completes the proof of (6.56). To prove (6.57) we again have t r i v i a l l y

(6.58)

Using t h e Cauchy-Schwarz inequality i n the form

161

LARGE ADDITIVE FUN(TTI0N.S where

we obtain

so t h a t (6.57) follows from (6.56), (6.58) and (6.60).

I t seems i n t e r e s t i n g t o a l s o i n v e s t i g a t e

(6.61) and (6.62)

and t o compare these suns with (6.43).

-

on t h e average considerably l a r g e r than greater than cz as

3:

+.

One feels t h a t B l ( n ) B(n)

, so

should be

t h a t (6.61) should be

f o r any constant c > 0

.

This follows e a s i l y

from Theorem 6.4.

As x

,

+.-

(6.63)

Proof.

x

.

Let

Let p1 < p 2


0 define

Then f

is additive and we have

since there is at least one prime p

between x/2

and x for x 2 3

.

The average order of an arithmetical function (as given by (6.4)) in m s t cases differs from the maximal order F(n)

, which may be defined

as the function F(n) satisfying

For example, from ( 2 . 2 ) and ( 2 . 3 ) , it is seen that loglogn is the average order of both w(n) and o(n)

(6.84)

which was used in (6.51).

n(n)

5

, while

logn/log 2

,

Furthermore for some C

>

0

,n

t

3

,

LARGE ADDITIVE FUNCTIONS

(6.85)

w(n)

Clogn/loglogn

s

and C l o g n / l o g l o g n

for n=nk

,

,

n(n) and w(n) a r e t h e functions

which implies t h a t t h e maximal order of logn/log2

169

respectively ( a t t a i n e d f o r n = Zk

th e product of t h e first k

md

(6.84) follows tri-

primes).

v i a l l y from a

n = p1

.

-I-. .+a

1

k

2"n)

while (6.85) follows from

u2(d) s d ( n ) s e x p ( C l o g n / l o g l o g n )

(6.86)

d ( n ) i s exp(Clogn/loglogn)

which a l s o shows t h a t t h e maximal order of

bince the maximal order i s a t t a i n e d again f o r n = n k first that if

primes).

k

n

p:'.

..p:

for all p

is fixed.

and a

t h e product of t h e

is t h e canonical decomposition of

n 6>0

,

To see t h a t t h e l a s t inequality i n (6.86) holds note

d o =k l az. t l where

,

6

We now have

, and

thus

do 6 n

i=l

a.6

pi

z

n

, then

'

( a t 1)/pa6 s 1 f o r p z 2"'

and

J.M. DE KONINCK AND A. IVIe

170

The desired inequality in the form

(6.87)

then follows for 6 = ((1

t

~ / 2 )log 2) /log log n

.

A more detailed account of

this subject may be found in Knopfmacher C11, Chapter 5. The inequality (6.23) is obvious from Stirling's formula for the gannna function if k is large, but it is also easily obtained by mathematical induction. For k = 6 we have 36 = 729

>

720

6!

, and the induc-

tion hypothesis yields

2;

(k+l)!(l+l/k)k

>

,

(k+l)!

since

In Definition 6.1, h ( x ) is a slowly oscillating function for 0< y < 1

.

It would be of interest to investigate

l'l/f(n) n%

(as well as F and F1 as given by (6.7) and (6.8))

, where

where

h(x)

is a

general non-decreasing slowly oscillating function, or even if h ( s ) = x%(x) where u > 0 and L ( x ) is slowly oscillating.

,

LAFGE ADDITIVE FUNCTIONS

171

The formulae (6.12) and (6.13) a r e not d i f f i c u l t t o prove.

From

the prime number theorem we obtain by p a r t i a l summation

(6.88)

P“Y c p

-

y2+ 0 2logy

1

- 5

Ilog2,

’

which gives

(6.89)

since

1 Urn2

rnsx

=

~ ( 2 )t 0(1/x) = n 2 / 6

t

0(1/x)

.

The proof of (6.13) is similar, and one can also prove

(6.90)

One of the few papers i n which B

and B

a r e investigated i s t h a t

of K. Alladi and P. Erdos C11, which contains many interesting r e s u l t s including proofs of (6.12) and (6.13).

For instance they prove t h a t f o r a

J.M. JE KONINCK AND A. IVId

172

fixed integer rnrl

,

as x-tm

, which

of n

Here Pi(n) (i = 1,.

n

.

, and

ple of

k,>O

reveals the connection between B

. . ,rn)

denotes the i-th largest prime factor of

is a constant depending on rn

c ( 1 t Urn)

.

, which is a rational m u l t i -

They also prove

1 (B(n) -

(6.92)

and large prime factars

nsx

B(n)) = x l o g l o g x t + ( x )

,

by noting t h a t

=

1 pcx/p21 P2”

t

c

pCx/p31 p3sx

t

...=x l o g l o g x t o ( z )

,

since

and

Further r e s u l t s concerning

B(n) and B(n) may be found i n the

paper of K. Alladi C11. There is an extensive l i t e r a t u r e about

LARGE ADDITIVE

One of t h e well-known estimates of

173

FUNCTIONS

Y(x,y) (somewhat weaker than (6.31))

can be found i n Prachar C11, Chapter 5. An estimate sharper than (6.31) can be found i n de Bruijn C3l. In (6.28) we used the estimate

since

because xo i s fixed. Note from (6.12) t h a t

B(n) = o ( n ) f o r almost a l l n

a

so t h a t

following the proof of Theorem 6.4 we may replace (6.63) with the sharper result

This Page Intentionally Left Blank

CHAPTER 7 RECIPROCALS IN SHORT INTERVALS

91. Introduction This chapter is concerned with the study of the sums

(7.1) where f belongs to a certain class of non-negative, integer-valued arithmetical functions. In '(7.1),

n belongs to the "short" interval ( x , t~ h l

where (as is customary in such problems) "short" means that h = O ( X ) z+-

.

,

, as

A n extensive literature concerning

(7.2) for various arithmetical functions f (or classes of functions) already exists. In addition to possessing an intrinsic interest, asymptotic formulae for (7.1) and ( 7 . 2 ) often lead to inequalities of the type

(7.3) where an is an increasing sequence of positive integers with interesting number-theoretic properties (such as a sequence of primes, a sequence of integers representable as a sum of two squares, etc.). approach to the estimate 175

Thus the classical

176

J.M. DE KONINCK AND A. IVIC

(7.4) where p ,

is the n-th prime, is based on establishing the asymptotic for-

mula

+ ( r th) - +(r)=

(7.5)

1

A(,)

x<nsxth

= (1 t o ( 1 ) ) h

,

as h-+m and c

(7.6)

h z x ,

whence (7.4) follows by taking x = pn

.

Our approach to the estimation of the sum ( 7 . 1 ) is based on the general method developed in Chapter 2 . We first estimate

(7.7)

.

We

where the error term we shall obtain will be uniform for IzI

5

shall then integrate this estimate over

, where

E(Z)

z

from

~ ( r to )

1

1

will be a suitably chosen function satisfying

lim

m

E(Z)

= 0

.

This procedure is particularly well-adapted to functions f for which z f ( n )

is generated by Dirichlet series of the form

RECIPROCALS IN SHORT INTERVALS where IzI 5

177

, considered as a function of s , is absolutely convergent on 1 , and Res = u 2 uo , for some uo < 1 . By an heuristic approach G

one may give an estimate for (7.1) for those functions f for which an asymptotic estimate of one would expect, for h

l’l/f(n)

n a

is known.

~ ( x )as

I+-

From Theorem 2.5, for example,

,

(7.9) but we are unable t o give a lower bound for h = h ( x ) such that (7.9) holds

82.

A n asymptotic formula for z f ( n )

in short intervals

We now prove Theorem 7.1, which will enable us to derive an asymptotic formula for the sum (7.1) whenever f belongs to a certain class of multiplicative or additive functions satisfying (7.8). Theorem 7.1

Let f ( n ) be a non-negative integer-valued arithme-

tical function such that (a) f ( n ) is multiplicative and f ( p )

(b) f ( n )

is additive and f(p)

If h = o ( z ) as

I+-

0

1 for all primes p for all primes p

, then uniformly on

IzI 5

,or

.

1

(7.11)

and a

(S

346/1067) is a constant for which the asymptotic formula

178

J . M . DE KONINCK AND A. I V I d

holds. Proof. We first assume that (a) holds. Let g,(n) by (7.11).

Then, for

If n

= pm

IzI 2

1 and for all n

, where (p,m)

f and the fact that f ( p )

1

be defined

,

, then by the multiplicativity of

1 we obtain

This implies

where c(n) is a multiplicative function such that 0

(7.16)

c(n)

if there is a p

such that p I In

,

= 2w (n)

otherwise.

If (b) holds, then (7.15) and (7.16) are still true, since g,(n)

179

RECIPROCALS IN SHORT INTERVALS

is multiplicative whenever f is additive. Moreover

p(1) zf(P)

gz(p)

For Res

>

1/2

t p(p)

= z o - zo

0

.

,

P

n=l

where (7.18)

is a Dirichlet series which is absolutely convergent for Re s > 1/3 the rough estimate

1 d(n)

n a


0

.

SY

We split the range n

i

SY

such that

Then

. x1/2 in (7.75) into O(1ogx)

intervals

of the form (IV,ZiVI and then use the above lemna with F ( t ) = zt-' X = IV

,Y

2117

, and

E

Z X I V - ~ to obtain (7.71).

,

A entirely elementary

proof of the same result (with log2x instead of logx)

can be found

in Vinogradov C11. One could also obtain (7.27) from a general convolution theorem of Tull 111, but our derivation, though longer, is self-complete.

and may be established by integration by parts (for more details concerning (7.34) see Chapter 8). To obtain the leading term in (7.48) one may use the Perron inver-

sion formula

where the prime

'

indicates that u ( n ) is replaced by u ( n ) / 2 whenever

x is an integer. When one moves the line of integration, the leading term in (7.48) is then simply the sum of residues at and

s

s=

1

, s = 1/2 ,

1/3 respectively. The best approach to the estimation of the

RECIPROCALS IN SHORT INTERVALS

199

error term in (7.48) consists again in transforming it into sums involving the function $(t) = t

-

Ctl - 1/2

is due to Srinivasan C11.

formula for

1a(n)

n-

.

The best result

This is also the error term in the asymptotic

(see Notes, Chapter 1).

The more general divisor

problems, such as the estimation of

1 1 ,

a b mn%

(1 5 a < b

, a , b integers) can also make use of estimates involving

(see Richert C 1 I) .

$(t)

For qntl - qn , the difference between consecutive square-free numbers, see Richert

C21,

where he proves

Further slight improvements were obtained by Rankin [ T I and Schmidt C11. Their basic idea is to investigate the range of h for which

1 p2(n) x

0 that for every

(8.21)

where a(n)

.

be the set of all positive mltiplicati-

ve functions f such that for some a 5

when ( k , Z ) = 1 or k

& .

Definition 8.2. Let

formly for -l/a

Then for certain functions

, and

ht(n)


0

is the constant appearing i n Definition 3.2.

thus obtain for some suitable C > 0

since , t r i v i a l l y

We

21 5

R E S T R I f l I O N TO PARTICULAR SEQUENCES OF INTEGERS

Before proceeding t o estimate (8.17) f o r c e r t a i n functions f E %,! we give a general i d e n t i t y f o r the generating function of any multiplica-

t i v e function f r e s t r i c t e d t o a congruence class f o r which t h e abscissa m

of convergence

u

1 f ( n ) n-' n=l

of

0

is f i n i t e .

Let f be a multiplicative function such that

Lemna 8.2. m

1 f(n)

converges absolutely f o r Res > u

n-'

then

n=l

m

n=l

, k'

where d = (k,Z) modk'

k/d

, I'

, and

= Z/d

the x's a r e t h e characters

. Proof.

If

(k,Z) = k (k, I )

becomes (8.3), while if

, then

n i l @ ) means n-O(k)

1 we have

d=1

, and

, and

(8.22)

(8.22) follows

from (8.14). Lemna 8.3. Res>l

and - l / a

Let f E 5

t

5

0

and l e t

( k , 2) = 1 o r k

.

Then f o r

,

216

J.M. DE KONINCK

AND A. IVId

where (8.24)

F(s,k,t)

p

=

t...)

-2a

2

t Y L - d mx t 1 ( P k'

cf(p)p

t

t . . .)

-1

P 2s

PS

-(atl)a t

at1 t

where d = ( k , l ) mod k

-1

k ' = k/d

, Z'

= Z/d

(f@

P(

and the

t...)

a) tP1 ) s

,

x's are t h e characters

. Since f i s positive and -l/a

Proof. 0 s ( f ( n )n-a)t

I,

1

.

t

5

0

we have t h a t

Thus the proof follows immediately from Lemna 8.2

with f ( n ) replaced by

(.f(n) n-")

I,

Cf(n)n-a)t

,

since the non-negativity of

implies the non-vanishing of t h e s e r i e s

217

RESTRICTION TO PARTICULAR SEQUENCES OF INTEGERS f o r Re s

2

1 , and assures absolute and uniform convergence f o r

of t h e Dirichlet s e r i e s generated by

Re s > 1

( f ( n ) n-")*, - l / a I t 2 0

.

We a r e now ready t o employ our basic method which was outlined i n Chapter 2 and was used i n Chapter 3 f o r summing reciprocals of logarithms of multiplicative functions.

1

mula f o r

We shall f i r s t e s t a b l i s h an asymptotic for-

, where f

belongs t o a subclass of

Let

&

ft(n)

&x, n Z ( k )

&,

, defi-

ned by Definition 8.3.

(8.25)

df(p)p-a)t

E

, where

[-A,O]

We c l e a r l y have

, then

and some 6

>

0

1 t O(p-&)

i n a r b i t r a r y but fixed, and

A >0

except possibly f o r O(xE)

fE

be t h e class of functions be-

such t h a t f o r every prime p

longing t o

for t

f

integers

nsr

5

4.

u

5

f o r which f ( n ) = 1 , In order t o see t h i s , l e t

from (3.4) we have

1- p - 8 s (f(p) p - a p s (1 - p - B ) - 1 = 1 O ( f 8 )

,

which implies (f(pj p T t =

{

(fb)P -"1l l q t / y

=

(l+O@-B))t/y = l t O ( p - 8 )

since t belongs t o a fixed i n t e r v a l .

From ( 3 . 4 ) we have f ( n )

, >>

n

a/2

.

21 8

J . M . DE KONINCK AND A. IVIi

Thus, f e

h1.

Similarly, using (3.20),

L e m 8.4. k

Let f~

.&,

and - l / a

&

5

A1 ,

5

t

5

0

.

If (k,Z) = 1 o r

, then

(8.27)

1

ft(n) = G(l,k,t) zattl attl nsx n-Z(k)

atG(l,k,t) + R ( 0 ) attl

where G is defined by

where d

(8.29)

(k,Z)

,k

k'ld

, and o(z l t a t - p 1

R(a,t)

uniformly in t for some fixed

p >

0

.

Proof. From the definition of ht(n) we have

(8.30)

which in view of (8.23) implies

Y

219

RESTRICTION M PARTICULAR SEQUENCES OF 1NTEC;ERS

m

(8.32)

B(s,k,t)

and f o r every

E >

1 b(n,k,t) n-' n=l

,

0

1 b(n,k,t)

(8.33)

=

n-

,

O(Z~-~'€) t ) '~'Z(O

since uniformly i n t

To see that (8.34) holds, we note t h a t using (8.25) we have f o r Res>l

n(1+X ( p ) f S

+0(f6) p - s + X*(p) ( p - 2 Q f ( p 2 ) ) t p - 2 s +.

P

=

n

P

(1 - x@) p - s ) -l

L(s,x)

t O(p-6)

P

n (l+O(p-6)p-St...)

P where

r[ ( 1 - x@) p-) ( 1 t x(p> p - s

.. 1

L(s,x)

C(s,x>

>

p - s t . . .)

J . M . DE KONINCK

2 20

AND

A. IVId

and

is regular f o r Re s > max (1- 6,1/2) for X = X

1

and k

, which

implies then (8.34), since

fixed,

A l l the products i n (8.24) of the type

are non-vanishing f o r Re s = 1 a regular function of

s

.

This means t h a t i n (8.24)

f o r Res > 1- p f o r some fixed

F(s,k,t) p >

0 , Using

a convolution argument we then i n f e r from (8.31) and (8.33) that

The above equation gives

(8.36)

is

RESTRICTION TO PARTICULAR SEQUENCES OF INTEGERS

I t i s now c l e a r why a condition such as (8.21) was needed. recalling t h a t from Theorem 1.4 we have f o r every

E >

221

Indeed

0

we obtain from (8.21)

f o r some

E~

> 0

.

Using p a r t i a l sumnation t o estimate

we obtain from (8.36)

which i n view of remarks made e a r l i e r gives

(8.40)

1

n ~ xn-2 , (k)

( f ( n ) n-’)*

= xG(lyk,t)

t

0(x1-’)

.

P a r t i a l sumnation f i n a l l y gives (8.29) f o r some fixed p > 0 which is not necessarily the s a m a t each stage of the proof.

J.M. DE KONINCK AND A. IVId

222

We are now ready to establish an asymptotic formula f o r the general

sum (8.17). Theorem 8.3.

Let f e

A1 , and l e t

1 or k

(k.,Z)

.

For an

arbitrary but fixed integer M z l we have

n-Z (k) where al = l / k

, and more

generally

a

(-1)

j

where E ( t ) = G(l,k,t)/(at

t

1)

j-1

E

(j-1)

(0)

, G(l,k,t)

,

being defined by the function

appearingin the statement of Lemma 8.4. Proof.

We make use of (8.26), which bounds f ( n ) away from unity,

except f o r O(xE) integers n s x nsx

satisfying f ( n ) 2 2

. Since

.

Let

c'

denote summation over those

f ( n ) >> naI2

2

for n z n

0

, we

for c > a

Since f ( n )

>>

na/'

for a l l n

, and

f(n) 2 2

for n z n o

,

have

RESTRICTION TO PARTICULAR SEQUENCES OF INTEGERS

1 f ( n ) dt= n~~:,nZ(k)

(8.44) -l/c

1

( E ( t ) xa t t l t a t E ( t ) +O(xl t a t - p

223

1) d t ,

-l/c

where E ( t ) = G(l,k,t)/(at t 1 )

.

Integration by p a r t s gives

Since E ( M ) ( t ) xat

O(1)

for t E Cl/c,Ol

, the

last above integral

is bounded, and, moreover, f o r O s i S M we have

X

di)(-l/c) x -a/c


1 / 2 any fixed A

>

0

the r e s t r i c t i o n m

.

(9.13)

Iz

I

5 A

for

Now using Lemma 2 . 1 (where i t may be e a s i l y shown that

It I

1 Ibt(n) I n - l log3 2n

n=l

and

2

1 may be replaced by

It I s B

is uniformly bounded f o r

It I

if 5 B)

we f i n d

J.M. DE KONINCK AND A. IVI6

234 where, uniformly for

Iz(

s 1

,

R ( t , z )