Area, lattice points, and exponential sums

LONDON MATHEMATICAL SOCIETY MONOGRAPHS NEW SERIES

Series Editors

H. G. Dales Peter M. Neumann

LONDON MATHEMATICAL SOCIETY MONOGRAPHS NEW SERIES Previous volumes of the LMS Monographs were published by Academic Press, to whom all enquiries should be addressed. Volumes in the New Series will be published by Oxford University Press throughout the world.

NEW SERIES 1. Diophantine Inequalities R. C. Baker 2. The Schur Multiplier Gregory Karpilovsky 3. Existentially Closed Groups Graham Highman and Elizabeth Scott 4. The Asymptotic Solution of Linear Differential Systems M. S. P. Eastham 5. The Restricted Bumside Problem Michael Vaughan-Lee 6. Pluripotential Theory Maciej Klimek 7. Free Lie Algebras Christopher Reutenauer 8. The Restricted Burnside Problem 2nd edition Michael Vaughan-Lee 9. The Geometry of Topological Stability Andrew du Plessis and Terry Wall

10. Spectral Decompositions and Analytic Sheaves J. Eschmeier and M. Putinar 11. An Atlas of Brauer Characters C. Jansen, K. Lux, R. Parker, and R. Wilson 12. Fundamentals of Semigroup Theory John M. Howie 13. Area, Lattice Points, and Exponential Sums M. N. Huxley 14. Super-Real Fields H. G. Dales and W. H. Woodin

15. Integrability, Self-Duality, and Twistor Theory L. Mason and N. M. J. Woodhouse 16. Categories of Symmetries and Infinite-Dimensional Groups Yu. A. Neretin

Area, Lattice Points, and Exponential Sums M. N. Huxley College of Cardiff University of Wales

CLARENDON PRESS 1996

OXFORD

Oxford University Press, Walton Street, Oxford OX2 6DP Oxford New York

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Published in the United States by Oxford University Press Inc., New York © M. N. Huxley, 1996

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Library of Congress Cataloging in Publication Data Huxley, M. N. (Martin Neil) Area, lattice points, and exponential sums/M. N. Huxley. (London Mathematical Society monographs; new ser., 13) Includes bibliographical references and index. 1. Exponential sums. 2. Lattice theory. 3. Functions, Zeta. I. Title. II. Series: London Mathematical Society monographs; new ser., no. 13. 1996 512'.73--dc20 QA246.7.H89 95-38370

ISBN 0 19 853466 3 Typeset by Technical Typesetting Ireland Printed in Great Britain by Biddies Ltd., Guildford, Surrey

To Trish, Nicholas, and Clare

Contents

Notation

xi

Introduction

1

Part I Elementary methods The rational line 1.1 Height 1.2 The Farey sequence 1.3 Lattices and the modular group 1.4 Uniform distribution 1.5 Approximating a given real number 1.6 Uniform Diophantine approximation

1.

Polygons and area 2.1 Counting squares 2.2 Jarnik's polygon 2.3 The discrepancy of a polygon 2.4 Fitting a polygon to a smooth curve 2.

The integer points close to a curve 3.1 Introduction 3.2 The reduction step 3.3 The iteration 3.4 Swinnerton-Dyer's method: the convex polygon 3.5 Swinnerton-Dyer's method: counting quadruplets 3.6 Expanding the result 3.7 Points on the curve 3.

The rational points close to a curve 4.1 Major and minor sides of the polygon 4.2 Duality 4.3 A linear form often near an integer 4.

Part II The Bombieri-Iwaniec method Analytic lemmas 5.1 Bounds for exponential integrals 5.

87 87

Contents

viii

5.2 Partial summation for exponential sums 5.3 Rounding error sums 5.4 Poisson summation 5.5 Evaluating exponential integrals 5.6 Bilinear and mean square bounds

89 93 98 104 114

Mean value results 6.1 The simple exponential sum 6.2 The lattice point discrepancy 6.3 The mean square discrepancy

126 126 128

The simple exponential sum 7.1 Motivation 7.2 Major and minor arcs 7.3 Poisson summation 7.4 The large sieve on the minor arcs 7.5 A preliminary calculation 7.6 Long major arcs

142

8.

The exponential sum for the lattice point problem 8.1 Major and minor arcs 8.2 The major arcs estimate 8.3 Poisson summation on the minor arcs 8.4 The large sieve on the minor arcs

167 167 170 172 178

Exponential sums with a difference 9.1 Major and minor arcs 9.2 The major arcs estimate 9.3 Poisson summation on the minor arcs 9.4 The large sieve on the minor arcs

185 185 187 188 193

6.

7.

9.

Exponential sums with modular form coefficients 10.1 Modular forms 10.2 The Wilton summation formula 10.3 Farey arcs 10.4 Wilton summation on the Farey arcs 10.5 A large sieve on the Farey arcs 10.6 Towards a mean square result 10.7 Jutila's third method 10.

135

142 144 149 152 158 160

197 197

200 208 210 213 220 225

Part III The First Spacing Problem: `integer' vectors The ruled surface method 11.1 Preparation: divisor functions 11.2 Families of solutions 11.3 A comparison argument 11.

235 235 241 249

Contents

ix

The Hardy-Littlewood method 12.1 Integrals that count 12.2 The minor arcs 12.3 The major arcs 12.4 Extrapolation

255

The First Spacing Problem for the double sum 13.1 Families of solutions 13.2 Good and bad families 13.3 The problem with a perturbing term

272

12

13.

255 259 262 270

272 278 283

Part IV The Second Spacing Problem: `rational' vectors The First and Second Conditions 14.1 The Coincidence Conditions 14.2 Magic matrices 14.3 The Second Condition 14.4 A family of sums

287

Consecutive minor arcs 15.1 Parametrizing rational points 15.2 Coincidence over a short interval 15.3 Linearizing the Fourth Condition 15.4 Coincidence over a long interval 15.5 Extending the Taylor series

307

The Third and Fourth Conditions 16.1 Counting coincident pairs of minor arcs 16.2 Sums with congruence conditions 16.3 Eliminating the centres of the arcs

329

14.

15.

16.

287 289 295 300

307 311 315 320 323

329 336 339

Part V Results and applications Exponential sum theorems 17.1 The simple exponential sum 17.2 Exponential sums with a parameter 17.3 A congruence family of sums 17.4 Sums with T large

349

Lattice points and area 18.1 Exponential sums 18.2 Integer points and rounding error 18.3 Lattice points inside a closed curve 18.4 A family of lattice point problems 18.5 Rounding error and integration

372

17.

18.

349 357 361 363

372 377 384 393 397

Contents

x

Further results 19.1 Exponential sums with a difference 19.2 A major arc estimate 19.3 Exponential sums with a large second derivative

406

Sums with modular form coefficients 20.1 Exponential sums 20.2 Mean value theorems 20.3 The modular form L-function

423

21. Applications to the Riemann zeta function

438

21.1 Introduction 21.2 The order of magnitude in the critical strip 21.3 The mean square 21.4 Gaps between zeros 21.5 The twelfth-power moment

438 441 443 447 451

22. An application to number theory: prime integer points 22.1 Preparation 22.2 Type I double sums 22.3 Type II double sums 22.4 Prime numbers in a smooth sequence

452

19.

20.

406 413 415

423 429 435

452 456 459 463

Part VI Related work and further ideas 23. Related work

467

23.1 Integer points close to a curve 23.2 The Hardy-Littlewood method 23.3 Other Farey arc arguments 23.4 Higher dimensions 23.5 Exponential sums with monomials

467 470 471 473 475

Further ideas 24.1 Comments on the method 24.2 Subdivision without absolute values

478

References

484

Index

491

24.

478 479

Notation GENERAL The lower case letters a to w denote integers unless stated otherwise. The letter p always denotes a prime number, and q always denotes a positive integer, usually the denominator of a rational number. The exceptions are as follows:

dy/dx etc., are derivatives. e and i are used for the usual mathematical constants when these occur. r, s, and t are used for variables where the notation is traditional: r for the

polar coordinate; s for arc length and for an exponent, and in Fourier transform formulae as in Dirichlet series; t for a real variable, usually `time', sometimes the imaginary part of a complex exponent s = o, + it; u and v are sometimes used for variables. x and y are always real variables, and z is a complex variable. Upper case letters are used for sets, matrices, and for real numbers (which may be integers). Coefficients (functions of one or more integer variables) are

usually denoted a(... ), b(... ), or c(... ). Functions of one or more real or complex variables are e(... ), f (... ), F(... ), g(... ), G(... ), h(... ), H(... ), k(... ), or K(...) and sometimes u(... ), v(... ), and w(... ). Greek letters are used for real and complex numbers and variables.

Theorems and lemmas are numbered uniquely. Theorem 18.3.2 is the second result in Chapter 18, Section 18.3. The equations in its proof are numbered (18.3.7) onwards, indicating that they occur in Section 18.3 of Chapter 18.

NUMBER-THEORETIC NOTATION The relation m I n (`m divides n (without remainder)') between m and n

means that m is positive and that n is a multiple of m. The relation a = /3(mod m) (`a is congruent to /3 modulo m') between real numbers, usually integers, means that a - /3 is an integer n and that m I n. For integers m and n, (m, n) denotes the highest common factor (defined for m and n not both zero) and [m, n] denotes the lowest common multiple (defined for m and n both non-zero).

The usual functions dr(n), µ(n), cp(n), and o,(n) of number theory are defined as follows:

Notation

xii

The number of ways of writing n as a product of r positive integers is called dr(n); we write d(n) for d2(n), the number of factors of n. The Mobius function µ(n) is zero if n has a repeated prime factor, 1 if n = 1 or if n has an even number of prime factors, all distinct, and is - 1 if n has an odd number of prime factors, all distinct. The Euler function cp(n) counts the number of expressions 1/n, 2/n,..., n/n which are in their lowest terms. The sum of the factors of n is called v(n). For a real number x, [x] is the integer part of x, the unique integer n in x - 1 < n <x, [[x]] is the nearest integer to x, the unique integer in x - z < n5 x + 2(W) is x - [[x]], depending only on x modulo one, and 114 is the

distance of x from the nearest integer, the absolute value of ((x)). The rounding error or row-of-teeth function is

p(x) = [x] -x+ i = - ((x - 2)). ANALYTIC NOTATION We write

e(x) = exp27rix = exp2iri((x)), a complex exponential that maps the reals modulo one to the unit circle. We use the following order-of-magnitude notation. The Vinogradov symbol

F 0, is complicated, partly because the argument itself is complicated, and partly because the two main parameters H and N are each chosen to equalize two or more terms in the upper bound. There is even some moonshine: the Fourier integral in the last Poisson

summation is an incomplete Bessel function integral. For modular form coefficients it has integral order, for the lattice point problem it has order one-half, and for the simple exponential sum, order one-third. Bessel functions of integral order occur in the symmetric analytic method, but only for the special curves, the ellipse and hyperbola. Bessel functions arise in the representations of matrix groups, but order one-third is unexpected. This book is accessible to graduate students beginning research. The reader is assumed to know some real and complex analysis, and Fourier theory up to the function (sin 7rx)/Trx. Some ideas from plane geometry are used, such as the radius of curvature, and the duality of points and lines. For the applications to modular forms, prime numbers, and the Riemann zeta function, standard theorems are quoted. No special knowledge of number theory is required. All technical lemmas on exponential integrals or on the average number of factors of integers that are needed are proved. This book can be read as an introduction to research in analytic number theory. There

are no set exercises. The researcher should keep asking: `What is this about?'; `How is it done?'; `Why is it done this way?'; `Can't you use instead ... ?'; `Isn't this the same as ... T.

Introduction

4

The expert will find the components of the new method presented in one

place for the first time, with suggestions as to how it relates to other approaches, and what direction it might take in the future. The results for sums in one variable in Chapters 17, 18 and 19 have been summarized for ease of reference as tables of exponents. Chapter 5 collects the most useful lemmas on exponential sums and integrals of one variable. A quick impression of the new method may be taken by reading Chapter 7 on the simple exponential sum, and Chapters 11 and 14 which begin the discussion of the two spacing problems. The list of references includes all the recent work (to 1993), and all the sources consulted in preparing this book, whether cited in the text or not. Thanks to Voronoi, van der Corput, and I. M. Vinogradov, without whom twentieth-century number theory would not be the same; to Davenport, a better teacher than I will ever be; to Bombieri, Hooley, Rankin, Selberg, and Swinnerton-Dyer for help and inspiration; to K. Ramachandra and the Tata Institute of Fundamental Research for the opportunity to work on this book; and to S. W. Graham, D. R. Heath-Brown, A. Ivic, H. Iwaniec, M. Jutila, G. Kolesnik, E. Kratzel, H. L. Montgomery, Y. Motohashi, C. J. Mozzochi, W. G. Nowak, 0. Trifonov, N. Watt, and everyone else who supplied ideas and comments.

Part I Elementary methods,

1

The rational line 1.1

HEIGHT

The rational numbers form a countable set, dense in the real numbers. They are set in the real line like stars in the sky `to illuminate the mystery of the

continuum' (a remark attributed to Borel). In any region, if magnified sufficiently, faint stars appear. To explain this bright and faint analogy, we need the concept of height.

A rational number a is uniquely represented as a fraction a/q in its lowest terms, that is, by a pair of integers a and q, with q positive and the highest common factor (a, q) equal to one. The height is defined by

H(a) = max(IaJ, q) There are three rational numbers of height one: 0, 1 and - 1 (unless infinity is counted as 1/0), and for each H z 2, the number of fractions of height H is 4cp(H), where cp(H) is Euler's function. The rational numbers are partially

ordered by height, and totally ordered when we put an ordering on the rationale of fixed height, for instance their ordering as real numbers. Each rational number of height H appears in the ordering in a position H

53+4F, (h-1)=2H2-2H+3. h=1

Thus the rational numbers are countable. We use the expressions `small rational number' and `large rational number' to mean rational numbers whose height is small or large. This usage agrees with the common expression `a small integer'. The small rationale are the

bright stars in the analogy, the large rationale are the faint stars, and the irrational numbers form the background darkness of the continuum. The height gives a metric on the positive rational numbers with distance function defined by

d(a, /3) = log H(a//3 ), so that the translation identity

d(ay,lay)=d(a,a), and the triangle inequality d(a, y) < d(a, /3) + d(13, y)

The rational line

8

hold. Any two distinct rationals are a distance at least log 2 apart. It is fairly easy to see that any set of n distinct positive rationals contains a pair which is at least log p apart, where p is the largest prime up to n. Zaharescu (1987), answering a question of R. L. Graham, has shown that for large n there is a

pair which is at least log n apart. This result is probably true for all n (to verify it for small n means checking a finite list of possible exceptional sets). This chapter is about the interaction of the height and the metric I a - '61

on the real line. In later chapters we shall use the structure of sets of rationals of bounded height close to a given real number. To a first approximation the height distance and the real distance are independent, but there is an analogue of gravitational lensing, in which the images of faint stars are displaced away from a nearby bright star.

1.2 THE FAREY SEQUENCE The Farey sequence of order Q (in the strict sense) consists of the rational numbers of height at most Q between 0 and 1, arranged in their order as real numbers. We usually use the Farey sequence in contexts where only the fractional part of a real number matters. The extended Farey sequence of order Q consists of all the rational numbers with denominator less than or

equal to Q, the lifting of the strict Farey sequence from R/71, the reals modulo integers, to the real numbers R. We write .9(Q) for the extended Farey sequence.

Although .9(Q) is a subsequence of .9(Q + 1), the numbering of terms differs, and consecutive terms of .9(Q) may be separated in .9(Q + 1). Farey

was one of the first to notice the two properties in the next lemma, and brought them to the attention of the great Cauchy. Lemma 1.2.1 (Basic properties of the Farey sequence)

(a) If e/r and f/s are consecutive terms in .9(Q), then

f/s - e/r = 1/rs,

ft - es = 1.

(b) If e/r and f/s are consecutive terms in .9(Q), then they remain consecutive in .A(R) for Q ::g R < r + s, but are separated in .9(r + s) by the Farey

mediant (e + f)/(r + s). Proof First we show that property (a) implies property (b). If fr - es = 1, and

a/q lies between e/r and f/s, then

a/q - e/r = (ar - eq)/qr > 1/qr, and

f/s - a/q = (fq - as)/qs Z 1/qs.

The Farey sequence

9

Adding, we have

1/rs = f/s - e/r>- 1/qr+ 1/qs, so that

q>-r+s. The Farey mediant (e + f)/(r + s) does lie between e/r and f/s. We find that

f/s - (e + f)/(r+s) =1/s(r+s), (e +f )I(r + s) - e/r = 1/r(r + s), and thus no other rational number with denominator r + s can lie within the open interval (e/r, f/s). This calculation verifies property (a) for the pairs e/r, (e + f)/(r + s) and (e + f)/(r + s), f/s, which are consecutive in .3(r + s), and implies that (e + f)/(r + s) is in its lowest terms.

By induction we see that we can construct the whole Farey sequence between 0 and 1 from the consecutive pair 0/1, 1/1 in .9(1) by repeatedly taking Farey mediants, so property (a) holds throughout. If 1/0 counts as a D rational, then 1/1 is itself the Farey mediant of 0/1 and 1/0. Definition We say that e/r and f/s are consecutive Farey fractions if they are consecutive fractions of some Farey sequence .9(Q), for instance with

Q=r+s- 1.

Our next lemma describes the set of rationale between two consecutive Farey fractions. Lemma 1.2.2 (Self-similarity) If e/r, f/s are consecutive Farey fractions, then for Q r + s the members of .9(Q) lying between e/r and f/s are the fractions (eu + ft)/(ru + st) with t z 1, u z 1, (t, u) =1 and ru + st s Q.

Proof By the inductive construction of the Farey sequence in the proof of Lemma 1.2.1, all the members of .9(Q) between e/r and f/s can be written as (eu + ft)/(ru + st). Conversely for t > 1, u z 1

f/s - (eu +ft)/(ru + st) =u/s(ru + st), (eu + ft)/(ru + st) - e/r = t/r(ru + st). If (t, u) = 1, then there are integers v and w with Iw

ul-1'

so that

ew+fv rw+sv

eu+ftl ru+st -

f l

s

elly r 11w

t1=1 u

implying that the fraction (eu + ft)/(ru + st) is in its lowest terms.

The rational line

10

We often need an upper bound for the number of members of .g(Q) in a short interval. Lemma 1.2.3 (Short interval upper bounds) Let I be an interval of length A. Then

E

150Q2+1,

a/q e.9(Q)n 1 1

- 2 and w is also an integer vector. Each rational number a/q corresponds to a pair of primitive integer vectors (q, a) and (-q, -a), and each primitive integer vector (x, y), except (0, ± 1), corresponds to a rational number y/x. An infinite gradient can be regarded as the rational number 1/0. We consider the parallelogram in Figure 1.2 with vertices 0 (0, 0), A (r, e), B (r + s, e + f) and C (s, f ). The area of the parallelogram is fr - es. Lemma 1.2.1 corresponds to the fact that when OABC has area one, then there is no other lattice point within the closed parallelogram OABC, and we can tile the plane with copies of OABC translated by every integer vector (m, n). The

lattice with basis vectors OA, OC has the same lattice points, but its lattice lines are in different directions. Changing the basis without changing the lattice points corresponds to an automorphism of the group of vectors given

by the matrix I sf

er).). The full automorphism group consists of all integer

matrices with determinant ±1. We want areas to be preserved, not reflected, so the order of the basis vectors matters, and we use matrices of determinant +1, the group SL(2, Also, ,since rational numbers form the gradients of

vectors, the matrix I -o (

_° I acts trivially. The group which affects the

FIG. 1.2

The rational line

12

gradients of vectors is PSL(2, Z), called the modular group, the quotient of SL(2, Z) by its centre ( ±I), represented by 2 X 2 integer matrices of determinant one, but with I _d) and I a d )) both representing the same group element. ` ` The matrices of the modular group are classified by the trace a + d, which is invariant under conjugacy. Since we can change signs, we take the trace to be non-negative. Matrices with trace less than two are called elliptic, those with trace two parabolic, elations, shears or transuections, those with trace

greater than two hyperbolic. The modular group acts as rigid motions on two-dimensional hyperbolic space. In this action a hyperbolic matrix is a rigid motion along an axis, an elliptic matrix is a rotation about a finite point, and a transvection is a rotation about a point at infinity. Conjugacy corresponds to changing the origin of coordinates. This geometric action is parametrized by the action

ay

as+b ca+d

on the complex numbers. The real axis parametrizes the directions at infinity. We interpret Lemma 1.2.2 as mapping the positive rationals t/u one-to-one

onto the rationals between e/r and f/s by the matrix I Sf er ). In Lemma 1.2.1 we have

r+s) = (s r)(0 1)' e+f e f e 1 0). (r +s r) (s r)(1 1 (s

The semigroup of matrices in SL(2, 71) with non-negative entries is generated

by the transvections T =

(10

1) and U = (1

o

). These matrices T and U

generate the full modular group. The usual generators are S and T, where

S= (0

-10) =T_1UT-i

1.4 UNIFORM DISTRIBUTION The uniform distribution of a sequence of numbers is a basic concept in analytic number theory. Dirichlet proved his prime number theorem by showing that the sequence of prime numbers was uniformly distributed among the cp(q) possible arithmetic progressions modulo q. In this book we

usually consider the sequence of real numbers x =f(n), where f(x) is a smooth function. We expect x,, to be uniformly distributed modulo one.

Uniform distribution

13

Definition A sequence of real numbers xn is uniformly distributed modulo one if, for every e > 0, however small, there is an N(e) such that, for every M>_ N(e), and every subinterval I of the unit interval [0, 1], then the number M(1) of integers n in 1, ... , M, whose fractional parts x,, - [ xn ] lie in I, differs from M times the length of the interval I by at most Me. We call the difference M(I) - M I I I the discrepancy of the interval I. Uniformity conditions are required when one approximate process or limit operation occurs inside another one. If g(x) is a continuous function satisfying the periodic condition g(x + 1) = g(x), then the uniform distribution of x,, modulo implies that

M M n=1

1

g(x)

(1.4.1) 0

as M tends to infinity. Periodic functions belong to Fourier theory, and Hermann Weyl showed that a sequence x,, is uniformly distributed if and only if (1.4.1) holds for the trigonometric functions sin 21rhx and cos 2.7rhx for

each integer h. We use the notation e(x) = exp 21r i x

for complex exponentials (when x is a rational a/q, then e(a/q) is a qth root of unity). In this notation the Weyl criterion becomes M

F, e(hx,,)->0 n=1

as M tends to infinity for each non-zero integer h. Weyl's criterion for uniform distribution will be proved in Lemma 5.3.3. In practice Weyl bequeathed us the principle that one may average away a variable, provided that certain exponential sums are small enough; what

bound is actually required for the exponential sum, that depends on the context. In this section we test the Weyl criterion for the rational numbers in (0, 1], enumerated as in Section 1.1.

Lemma 1.4.1 (Ramanujan's sum)

F e Aqq

a=1 (a,q)=1

clh clq

cµ(

where µ(n) is the Mobius function. Proof Since

E µ(d)=0 din

The rational line

14

if n is greater than one, and the factors of the highest common factor (a, q) are just the common factors of a and q, the sum required is q

E

q1

a=1 din

q/d

b=1

dlq

where we have written a = bd. 1 he sum over b is zero unless h is a multiple of q/d so, writing c = q/d, we get /

F, µ(d) d = E F, cµ(d) _ > cp q clh

dIq

q/dlh

clh clq

d

cd=q

1 ).

`

Lemma 1.4.2 (Weyl's sum for the rational numbers) Let Q be a positive integer, and let x1, ... , x v be the rational numbers of height at most Q in (0,11. Then

N= 2Q2) +O(QlogQ)=

3Q2

+O(QlogQ)

and for each integer h N

dM(Q/d),

E e(hxa) dlh

1

where M(x) = Ea

x

µ(n) is the sum function of the Mobius function.

Proof For the first assertion, by Lemma 1.4.1 with h = 0, Q

q=1

Q

q

N= >

1= E F, F, cµ(d)

F, a=1 (a,q)=1

q=1

c

d

cd=q

Q

_ E E d µ(d) _ E

µd d)

d5Q

q=1 dIq

Q

E

q

q=1 q = 0 (mod d)

p(d) d

d5Q Q2

°r°

E

2 d=1

2 1

Q2 + O(Q) d

u(d) + d

2

O

I µ(d)1

Q2

d=Q+1

d

2

+ Q X-

dsQ

I µ(d)I d

The error terms are O(Q log Q) in order of magnitude. The infinite series is

1/i(s) at s = 2, and (2) takes the special value ire/6.

Approximating a given real number

15

For the second assertion, the sum can be rearranged to give

E

q=1

F, a=1 (a,q)=1

F, EdjA

elah q

d

q=1 dlh

dlq

Ed dlh

µ(r). r5Q/d

Lemma 1.4.2 gives the Weyl sum in the case when the enumeration x1, ..., XN corresponds to the complete set of rationals of height at most Q. If N is such that xN has height Q, but so does xN+1, then we must add O(Q) to the results of Lemma 1.4.2 to correct for the terms omitted. Since I M(x)I <x, we have, for non-zero h, N

Q Q + 1 since the mediant is not in 9(Q). Similarly if x > (e + f)/(r + s), then we take a/q = f/s. 0 The first proof is a variation of Dirichlet's pigeon-hole principle, that if Q + 1 letters are delivered to Q persons, then somebody has at least two letters in their pigeon-hole. The pigeon-hole principle is used like compactness in analysis. The second proof can be made constructive by the continued fraction rule, which, given a real number x, constructs a sequence of rational numbers, alternately below and above x. If x is rational, then this algorithm finds the

Approximating a given real number

17

lowest-terms form of x, and then terminates. In particular, if x is given as a ratio m/n, then the continued fraction rule is simply Euclid's algorithm to find the highest common factor d = (m, n) and integers u, v with n m

ul =d.

In our notation r = m/d, q = n/d, and r/q, v/u are consecutive Farey fractions.

The continued fraction rule is an iteration. Let

ro=ao,

ao = [x},

qO = 1,

and, if x :k a0, then let

x1 = 1/(x - ao). Then 1

ro

0

qO

and

x=

rox1+l gox1 + 0

For the second step, let

a1= [xl],

(ql)

0)(

f qO

1

and, if x1 0 al, then let

x2 = 1/(x1 -a1). Then r1

ro

ql

qO

= 1.

and

x=

r1x2 + ro g1x2 + qO

For the general step, let

(rn) =rrn qn

I11 qn

1

rn-2

-1

qn-2

and, if xn # an, then let

xn+1 = l/(xn -a,,).

1 ( an)

The rational line

18

Lemma 1.5.2 (Continued fractions)

At each step in the iteration given above

we have

r_1

rn

qn- 1 qn

_

(-1)n,

(1.5.1)

and

rnxn+l

(1.5.2)

gnxn+l +qn-1 lies between rn/qn and rn_ /qn-1, which are consecutive fractions of A q,,). The fraction with the even suffix is the smaller of the two. If x is irrational, then the sequence r,,/qn tends to x with 1

1

gnqn+l

2n

Proof We see that I

rn-1

rn

rn-1

rn-2

qn-1

qn

qn-1

qn-2 I'

and the first assertion (1.5.1) follows by induction. Substituting

xn+1 -an+1 = 1/xn+2 in (1.5.2) gives

x=

(rnan+l +rn-1)xn+2+rn (gnan+l +qn-1)xn+2 +qn

proving (1.5.2) for all n >- 2 by induction. We find that rn-1

-x=

qn-1

(-1)nxn+1 qn-1(gnxn+l +qn-1)

and rn

(-1)nxn+1

qn

gn(gnxn+l + qn-1)

Both these differences have the same sign, which is that of (-1)", since xn + 1, q,, and qn _ 1 are positive. The fractions rn/qn and rn - 1/qn are consecutive (in the appropriate order) because of (1.5.1) and Lemma 1.2.1. For the third assertion we note that qn+1 ?qn +qn-1z2gn-1, gnqn+l ? 2gngn-1,

and by induction gnqn+l ? 2"gogl

? "2

Uniform Diophantine approximation

19

Since x lies between r,,/qn and

we have 1

The matrices rn-1 qn-1

rn

qnJ

which occur in the construction have determinant ± 1, and rn-1

r,,

qn-1

qn

=

rn-1 qn-1

1an)

1

q,, -2 I\0

rn-2 1Ta,,

_

l qn-1 (r 21

qn-2J

in the notation of Section 1.4. Similarly we have rn+1 qn+1

rn qn

_

rn - I

rn

qn-1

qn

1

(a,

0 1

_

n-1

- (qn-1

)s_1Ts. r

We note that, although S-1 = -S as matrices, in the action of the modular group PSL (2, Z), S-1 = S.

1.6 UNIFORM DIOPHANTINE APPROXIMATION Although the average gap between the fractions of .9'(Q) is in order of magnitude 1/Q2, there is a gap of length 1/Q between 0 and 1/Q, and there are long gaps about other small rational numbers. In Borel's analogy, a bright star is surrounded by an area in which the stars are fainter than usual, as if the images of nearby stars have been displaced away from the bright star. If criterion (A), accuracy of approximation, is the most important, then, given an accuracy S (with 0 < S < 1), we need a set of reference fractions such

that every real number is distant at most S from one of the reference fractions. One way to do this is to divide the real line into intervals of length at most 3, and to choose the smallest rational number (smallest in height) in each interval to be a reference fraction. If the extended Farey sequence .9'(Q) has SQ2 < 1, then no two fractions of 9(Q) fall into the same short interval, so that all the fractions of .9'(Q) are reference fractions. Some intervals will

give reference fractions with denominator greater than Q. Choosing the smallest rational in each interval is what is called a `greedy algorithm' in combinatorics. The following construction gives an explicit set of reference fractions. We

start with the Farey sequence .9'(Q) for Q = [2/1/S ] and refine it to a sequence of rationals at most S apart. If e/r and f/s are consecutive Farey fractions with

If/s-e/rI<S,

The rational line

20

then we leave the interval [e/r, f/s] alone, but if S < I f/s - e/rl = 1/rs, then we fill the gap. Suppose that r > s. We insert fractions e,

e+ft,

f

1

ri

r+st;

s

s(r+st;)

which are adjacent to f/s in their respective Farey sequences. We choose t; by to = 0, and for i z 1 r + st; = 2/isS + O(1), the sequence stopping at t; when 1

s(r+st;)

5 S.

We note that, since 2r - 1 z r + s > 2/15, we have

r+st;>rz1/,/S. We have enlarged the Farey sequence AQ) to a sequence of reference fractions. The gaps between consecutive fractions have lengths between 8/4 and S. Our next lemma, on the greedy construction and a variant of it, is taken from Huxley and Watt (1988). Lemma 1.6.1 (Uniform Diophantine approximation). Suppose that we are given a real interval J of length 0, divided into subintervals Jk of length at least S,

where 0 < 6 < 1. In each interval Jk we pick the rational number ak/qk with least denominator. Let Q be any positive integer. Then the inequality qk 5 Q holds for at most OQ2+1 values of k, and qk Z Q holds for at most 80

8

62Q2 + SQ values of k.

If we pick instead the rational number ak/qk in Jk with qk least, subject to qk z 1/3, then qk z Q holds for at most 240

12

S2Q2 + SQ

values of k.

Proof The first assertion is part of Lemma 1.2.3. Intervals for which qk > Q contain no fraction of .1(Q - 1), so they lie between some pair of consecutive

Uniform Diophantine approximation

21

fractions e/r and f/s of 9(Q - 1). Suppose that s is the smaller donominator; then s < r, r >- Q/2. If there are n values of k for which the interval Jk lies strictly betwen e/r and f/s, then

n6- Q that belong to a fraction f/s outside J. All other intervals Jk with qk >- Q belong to some fraction f/s within J. Thus by Lemma 1.2.3 we find that qk >- Q occurs for at most 2 2-+2 E 6Q ss2/sQ

E f

2 SsQ

8i

< S2 Q 2

+8

SQ

values of k. In the case when choices of ak/qk with qk < 1/6 are not allowed, we may suppose that Q2 z 4/6, or the result follows from the trivial bound L/S + 1

for the number of subintervals. We must add the number of intervals Jk containing a fraction a/q in 9(1/%6) (which must be unique) and no other member of 9(Q - 1). Then q 5 Q/2. Suppose that a/q lies between e/r and f/s in .9(Q - 1). Since the mediant (a + e)/(q + r) is not in F(Q - 1), we have q + r >- Q, so that r >- Q/2. Similarly we have s z Q/2. We deduce that

SSf/s - e/r= 1/qr+ 1/qs M1, then all points inside M1C are inside M2C. Thus K(M) and K(M) + L(M) are increasing functions of M. We show that the limit of K(M)/M2 exists as follows. Let n be a positive integer. Then each unit square inside the curve MC corresponds to n2 unit squares inside nMC, and each unit square cut by MC corresponds to n2 unit squares, some of which are cut by nMC. Thus we have K(nM) >_ n2K(M),

L(nM) 5 n2L(M).

If L(nM)/n2M2 tends to zero as n tends to infinity, then K(nM)/n2M2 tends to a limit, the area inside the curve. We show that the area exists when C is a convex curve, of length A say. The curve nMC has length nMA, and the x-coordinate changes by at most nMA/2 on the curve nMC. Hence at most nMA/2 + 1 lattice lines parallel to the y-axis are cut by the curve nMC (Fig. 2.3), since the curve has to go there and back again. Similarly there are at most nMA/2 + 1 lattice lines parallel to the x-axis which are cut by the curve.

As the curve nMC is described in the sense of t increasing, it passes

Counting squares

27

N

nM.U2 FIG. 2.3

through L(nM) lattice squares. If the curve never cuts a lattice line, then L(nM) = 1. If L(nM) > 1, then the curve nMC enters each new square by cutting a lattice line. Since nMC is a convex curve, it meets lattice lines in two distinct points, or possibly in a closed interval. In the case of a closed interval, we take the endpoints of the interval to be the two points in which the lattice line is cut by the curve; they may coincide. Then L(nM), the number of squares cut, is at most the number of times that the curve nMC cuts a lattice line in this sense, and L(nM) < 4(nMA/2 + 1) = 2nMA + 4.

We see that L(nM)/n2M2 tends to zero, and the common limit, A, of K(nM)/n2M2 and (K(nM)+L(nM))/n2M2 does exist. To show that the limit is independent of M, we use the fact that if n and r are integers with rM M1"d+E and a certain number of derivatives not vanishing. Bombieri and Pila (1989) took these ideas further, getting the bounds

R(M) > A2/3 >> A1/3.

Proof Let the vertices of the Jarnik curve be PI,..., PR. Let Cr be the circumcircle of the triangle Pr-1PrPr+1 (Pa is identified with PR, PR+1 with P1). The triangle Pr_1PrF.+I has area i by Lemma 1.2.1, and sides of length

between 1 and HV. Either Pi_1Pr or PrPi+1 is a vector with one entry at least H/2. Let Or be the angle 1 _1PrPr+1 (Fig. 2.6), K, be the length of Pr Pr+1, la'r be the length of Pr_1'r+1, and let pr be the radius of the circle Cr. By elementary geometry 1

2

1A'r

= Kr- 1 K,sln 6r,

sin Or e = 2Pr'

so that Pr = Kr-1 KrPA'r s Kr-1Kr(Kr-I + Kr) 5 4H3V.

The circles Cr and Cr+1 meet at P. and at Pr+1. We now take circles Dr of radius 2 pr, such that Cr and Dr touch internally at P. (Fig. 2.7). The circles

Dr and Dr+ 1 meet outside the circles Cr and Cr, 1, so that there is an intersection between P. and Pr+ I and outside the Jarnik polygon. We can

Dr+l

FIG. 2.7

The discrepancy of a polygon

33

join P and P,+ 1 by a smooth curve E, which has the same tangent and radius of curvature at the point P, as the circle D, and has the same tangent and radius of curvature at the point Pr+ 1 as the circle D,+ 1, and has radius of curvature at most 4HVV. When we subtract the equation of Dr from the equation of D,+ and change coordinates, then this corresponds to finding a 1

function g(x) with g(0), g'(0) and g"(0) all zero, g(x), g'(x) and g"(x) non-negative and bounded on 0 < x 51, and taking prescribed values at x = 1. The length of Er is less than the length of the polygonal path from P, to P,+ 1 formed by the tangents to the circles Cr and Cr, 1 at Pr and P,+ 1,

respectively. The angle between the tangent at Pr and the line P,P,+1 is equal to the angle P,Pr_1Pi+1, which is less than IT- Or. If H is large, then Ors 31x/4, the tangents cut at an angle greater than 7r/2, and the length of Er is Ar 5 (sec 0, + sec 0,.+1)K,:!,r+ 1) K, S 2K,'!1.

The Jarnik curve is formed from the curves Er, having continuous radius of curvature, and length

A= EA, 1, or if the equation has the form x = y, then we assume that

we treat y as the independent variable. Let E be the region between the lines x = t + 2, x = u + 2, y = v + i and y = ax + /3, a quadrilateral whose R, and Pn_1Pn, extended if necessary. The quadrilateral may be self-intersecting. We take area and sides lie along the lines Rn_1Qn_1, RnQn, Q,,

lattice points above the line y = v + i as counting positively, those below as counting negatively. Then the signed number of lattice points in D differs from the signed number of lattice points in E by at most two, and the area within D differs from the area within E by at most four, the difference being

contained within lattice squares containing Q,,-, and Q, and adjacent squares above or below these two. The area of E is 2 a-v-)dx=(u-t)

(a(t+ u+1) 2

1

t+

1

+/3-v-2Ji

and the number of lattice points is u

u

m=t+1

1+1

F, ([am+/3]-v)= L(p(am+/3)+am+/3-v-Z) U

_ L p(am+/3) t+1

+(u -t)

(a(t+u+1) 2

+/3) -v-Z

Polygons and area

38

Hence the signed number of lattice points in the pentagon D differs from the area of D by u

E p(am+/3)+O(1). r+1

We put together the bounds given by Lemma 2.3.2 for each such pentagon to obtain the error estimate in counting squares for a convex polygon. Theorem 2.3.3 Let D be the region inside a convex plane polygon P1P2 ... P. as follows. 1P,, (with P0 = PN) define the bound 1, For each edge

Let L be the length of the edge P1P,, and a be its gradient, with continued fraction a = ao + 1/(a1 + ), and convergents r,/q;. Let k be the largest integer with qk 5 L + 1. Then

B(Pi-1,

=ao+a1 + ... +ak + (L + 1)/qk

In this notation the discrepancy 0 of D, the difference between the number of integer points in D and the area of D, satisfies N

0
>-1

E B(u) - 0. Then the number of solutions of (3.1.1) with F(x) = G(x) R 0,

-1-1/r

if -K 28 (u1 + U2).

Since u1 and u2 are positive integers, u1u2 > (u1 + u2)/2, and

d21>(S(u1+u2)-1)2-1.

(3.4.4)

Swinnerton-Dyer's method: the convex polygon

55

We see that d21 (and similarly d;+1,i) cannot be negative. The integer points Pi, (mi, ni), close to the curve form a convex polygonal line. The gradients vi/ui form a decreasing sequence. No four vertices form a

parallelogram, so the differences m, - mi, n1 - ni together determine the points Pi and P,, we use this in the next section. We call PiPi+1 a minor side

of the polygon if neither P;_1 nor P;+2 lies on the straight line PiPi+1. A major side of the polygon consists of a maximal sequence Pi, Pi+ 1,

, Pi+, of

collinear consecutive vertices, along some line y = (ax + b)/q with a, b, q integers, and a/q in its lowest terms. By (3.4.2) the gradient a/q lies in an interval I with length j

AL+

AL

There are at most two intervals of x on which TFlMax+blS6.

(3.4.5)

q

The integer points Pi,..., Pi+, fall into one or other of these intervals. A proper major side is a maximal sequence Pi, Pi+ 1, ... , Pi +, of at least three collinear consecutive vertices on a line y = (ax + b)/q, lying within an interval of x on which (3.4.5) holds. The integers mi, ... , mi+r are in arithmetic progression with common difference q. We see from (3.4.3) that a proper major side has length at most K, with qt- 0 for which 2` > k1 > 2`-1

(3.5.10)

Here r takes at most log N/ log2 + 1 values. We treat the trailing triples similarly, according to the size of k3. Suppose that a1, a2, a3 and k1, k2, k3 are leading triples in the same class (3.5.10). From (3.5.7) and the analogous determinant formed with the k, we have c31(a2k1 - a1k2) = c21(a3k1- a1k3), c31(a2k3 - a3k2) = c32(a1k3 - a3k1). We see that the integer

e = a1k3 - a3k1

(3.5.11)

is divisible by

_

C31

C31

(C211 C31) ' (C31, C32)

_

C31

C31

(c21, C311 C32)

ac

Eliminating c21 from (3.5.8) and its analogue for the ki gives C32(C31+C32)(a2k2-a2k;)

1B

(4.1.7)

2 8'

and at most 2Q(2C0)1"3 M

(4.1.8)

values of r with ar+1

qr+ 1

ar+2

qr+2

ar+2

qr+2

ar

qr

ar+1

qr+l

ar

qr

1

>_

2grgr+lqr+2

(4.1.9)

2

for some e between 0 and M. Corollary

The number of rational points on the curve which can be written as

(a/q,b/q) with QSq 1/2grgr+lqk ? 1/8Q2gk Hence the determinant D.Jk has size I D;JkI

= 2g1gjqk(area PiPi Pk) > d/4.

In Lemma 4.1.2 we have either I80iikl z d/8,

so that (4.1.16) holds, or, for some e a, ai

q,

ak

qi

ai

qk

ak

qk

AF"(/M)

qi

a,

q,

2gIgjqk

d

8)

giving

a,

a;

ak

a1

ak

a1

qi

q,

qk

qt

qk

q,

d 32CiiQ3 '

which implies the remaining inequalities.

Lemma 4.1.7 (Separation of major sides)

j-i=dz2,

Let P,P1 be a major side, with aj a;

- - = A,

q1

q1

and equation lx + my + n = 0. Then a,+1 q1+ 1

-a

>_min

q1

F 2m

d

d

i

6 4S Q2' 1 6 AQ 2

(4.1.18)

and also aj+1

ai

q1+1

q1

->min

d

d

192 SQ2 ' 48C Q2

m ( 2S )

(4.1.19)

Proof Since the line contains d + 1 rational points with x values at least m/4Q2 apart, we have the inequalities A Z dm/4Q2,

d :!!g 4Q2A/m.

The rational points close to a curve

70

If the second possibility (4.1.17) of Lemma 4.1.6 holds, then

- a,12

(aj+i qt+i

q,

d

d2m

J Z 32C Q3A

128COQ5A2'

which gives (4.1.18). For the bound (4.1.19) we modify the proof of Lemma 4.1.6. Let e = [d/2] > d/3. By (4.1.15) of Lemma 4.1.5 we have as/q3 - ar/qr 5 AA'

either with r = i, s = i + e or with r = i + e, s = i + d. We argue as in Lemma 4.1.6 with i, j, k replaced by r, s, i + d + 1. Again we have

e 5 4Q2µ/m.

µ >_ em/4Q2,

(4.1.20)

Corresponding to (4.1.17) we have ar

ai+i ( qj+i

e

2

-, Z 32C Q3µ qr

elm

>

elm 512C2Q' '

128C OQSµ2

giving the second case of (4.1.19).

Lemma 4.1.8 (Vertical major sides) most 366MQ2 rational points.

O

The major sides with m = 0 contribute at

Proof In the case m = 0 we have (1, n) = 1, so x = -n/l, y = b/q with 1 1q .

Let q = kl. Then ly lies in an interval of length 281/Q, with denominator k ::g 2Q/l. By Lemma 1.2.3, there are between 3 and 281 4Q2 Q

+1 12

88Q + 1 5 128Q 1

1

choices for b/q for each fixed n/l. Also, n/l lies in an interval of length M, and 15 2Q, so by Lemma 1.2.3 again, vertical major sides contribute at most

Fl128Q 1

:!g 128Q(2QM+ 1) 5 366MQ2.

0

n/[

Lemma 4.1.9 (Good major sides) Each individual major side contributes at most 72

(2 S CQ3

(4.1.21)

m

rational points. For A z 1, the major sides with m0 0 and A 5 4A a

(CQ)

(4.1.22)

Major and minor sides of the polygon

71

contribute in total at most 144

2SCQ3

0

0

) + 192ASMQ2

(4 . 1 . 23)

rational points, and for B S 1/S the major sides with m >- 1/B8

(4.1.24)

588B1/2CSMQ2

(4.1.25)

contribute in total at most rational points.

Corollary The number of rational points on major sides is at most 606B112CSMQ2.

Proof We use the notation of Lemma 4.1.7. Since d >- 2, the number of rational points on a major side satisfies d + 1 5 3d/2, and by (4.1.20) in the proof of Lemma 4.1.7 we have

d < 3e S 12Q2µ/m,

(4.1.26)

which gives the first assertion (4.1.21). For the other bounds we adapt the argument of Lemma 4.1.3. We number the major sides satisfying (4.1.22) from

1 to K, and call their endpoints P,(k), Pj(k), with dk =j(k) - i(k). We put Xk = ai(k)/qi(k) Since A + 1) >- j(k) and i(k + 2) > j(k) + 1, the bound (4.1.18) of Lemma 4.1.7 gives xk+ 2 - xk ?

as(k)+1

a'(k)

dk

- qi(k) - 64A S Q2 qj(k)+1 2

Thus K-2

K-2

E dks64ASQ2 k=1

(xk+2-xk)5128ASMQ2. 1

The bound (4.1.23) follows when we use (4.1.26) for the final values k = K - 1

and K, and the inequality dk + 1 < 3dk/2. Similarly, we number the major sides satisfying (4.1.24), and define Xk likewise. By (4.1.19) of Lemma 4.1.7

xk+2 -xk z dk/96B112C8Q2. and by (4.1.26) 2 SCCQ3

d k 5 48BS

48S

2BCQ3 :!9 - 96B112CSMQ2

for each k, since OM2 Z 1. This gives the bound (4.1.25). The corollary

The rational points close to a curve

72

follows when we give B its maximum value 1/8 in (4.1.25), and add the bound of Lemma 4.1.8 for the number of rational points on vertical major

0

sides.

4.2 DUALITY We need the notion of the function g(y) complementary to f(x). It

is

convenient to have f(x) defined for all x, with f"(x)> 0, although we only

use values of x in the range 0 to M. The function F(x) is defined for 0 < x< 1, where it satisfies (4.1.4). Suppose that (4.1.4) holds on a range a - /3. This gives a function F(x) defined for all x, twice continuously differentiable, satisfying the inequality (4.1.4), and agreeing with the original definition for

0<x< 1. We continue to put f(x) = TF(x/M). The inverse function h(y) of f'(x) is defined for all y. We put

g(y) =yh(y) -f(h(y)), so that

g'(y) =h(y) +yh'(y) -f'(h(y))h'(y) =h(y), and

g"(y) =h'(y) = 1/f"(h(y)). We can easily check that f(x) is the function complementary to g(y). Figure 4.1 shows the relation between the complementary functions f(x)

and g(y) in the special case when f(x) vanishes at the origin. They are represented as areas bounded by the curve y =f'(x).

g(y)

fl FIG. 4.1

Duality

73

Lemma 4.2.1 (The dual curve) On a straight line lx + my + n = 0, there are at most two disjoint intervals with the property that every point on them has

l y -f(x)l5 S/Q.

(4.2.1)

If I is such an interval, containing two rational points Pi, Pj satisfying (4.1.1), with

A=ai ---aiqi q1

then min XE!

l

28

m

AQ

- + f'(x)

,

(4.2.2)

and

g(

l

n

m

m

2C3S2

S

Q+AA2Q2

(4.2.3)

Proof Since f"(x) > 0, the points x with

f(x)5 -(lx+n)/m+S/Q form an interval, which may be empty. The points with

f(x) S -(lx+n)/m-S/Q form a subinterval, which again may be empty. The difference of these sets is one or two intervals on the real line.

Let U, (xo, yo), be the point on the curve y = f(x) where the gradient is

-I/m. Then xo=h(-l/m), so that Yo + lxo/m + n/m = n/m -g(-l/m). In the one-interval case, the value xo must lie within the interval, so lxo

Yo+m-

+

n

m

S

S Q.

In the two-interval case, the value xo does not lie in either interval. We use the rational points Pi and P. By subtraction

f

so, for some

aj

ai

-f(gi

ll

+

l

a,

ai

m q1 qi between ai/qi and at/qj, we have, by the mean value qt

theorem, 1

MO +In This completes the proof in the one-interval case.

The rational points close to a curve

74

In the two-interval case, the point xo does not lie in either interval. A second application of the mean value theorem gives 28

A CIA-xols

Q.

lies between a./qi and a,/q1, whilst xo does not, then either ai/qi or a!/q, lies between l; and xo. Thus for P, (a/q, b/q), equal to either P, or Pj, Since

we have

la

-x

IS

2CS

AAQ'

V

and Taylor's theorem about xo gives

b qS6=

x +(q -x)

(4

2

for some q. Since f'(xo) _ -1/m, and la + mb + nq = 0, we see that lxo

I

YO +

nI

2C3S2

S

m + m 5Q+-z-

which completes the proof of the lemma.

Lemma 4.2.2 (Points close to the dual curve) The number of rational points on major sides is O(C2SMQ2 + C7/651/2A1/6MQ11/6 )

Proof For technical reasons we want an upper bound for the length A of a

major side (measured in the x-direction). If A exceeds the bound µ of Lemma 4.1.5, then we argue as in Lemma 4.1.7. Let the points on the major

side be Pi, ..., Pi+d. Then for e = [d/2] either Pi,..., Pite or Pi+el

) Pi+d

span a distance at most µ in the x-direction. Instead of considering the whole

major side, we take the shorter of the two halves, which contains at least (d + 1)/2 rational points, including the endpoints. Hence we can suppose that A 5 p., provided that we double the final estimate for the number of rational points. We use Lemma 4.1.8 for major sides with m = 0, and Lemma 4.1.9 with A = C for major sides with length A satisfying A 5 4S

(---) K

.

(4.2.4)

We use Lemma 4.1.9, with a value of B to be chosen, for major sides with m > 1/BS. Let N be the length of the interval taken by f'(x) for MS x5 2M, so that N< CAM. By (4.2.2) of Lemma 4.2.1, major sides for which (4.2.4) is false have 1/m within a distance AQ 1 ( 28 A

52

I\

J« 1i cM

2Cm AQ)

Duality

75

of a value of f'(x). Thus l/m lies in an interval of length M' 0, the major sides contribute

O I C19/9SMQ2I CQT I E+ C11/661/2A1/6MQ3/2log 1 J (QT)

= OI C10/36MQ2

E

+ (CA)1/3MQ)

(4.2.8)

rational points. The implied constants depend on e.

Proof We can assume that (4.2.9)

Q >_ 2C/e2,

since the result follows from Lemma 4.2.2 if (4.2.8) is false. As in Lemma 4.2.2, we use Lemmas 4.1.8 and 4.1.9 with A = C for major sides with m = 0, m >_ 1/BS or with (4.2.4) false. These cases give the terms of (4.2.7). Let N be the length of the interval taken by f'(x) for M 5 x 5 2M, as in Lemma 4.2.2. By (4.2.2) of Lemma 4.2.1, major sides for which (4.2.4) is false have 1/m within a distance 2S

eAM

1

AQ2

(2CmQ) < 4C 11

eN 4

of a value of f'(x), since N>_ OM/C> 1/C, and (4.2.9) holds. Thus 1/m lies in an interval of length

M'M. We write

0=T/M2. Let f(x) = TF(x/M), and let R be the number of points (x, y) which can be written as (a/q, b/q) with a, b, q integers, 0:5 x 5 M, Q:5 q < 2Q, and with

Iy-f(x)I=

b

-f(a

S

)I < Q 9 Then, as either M, Q, or T tends to infinity, we have the bounds R 0 on (a, 13). Let g(x) be real, and let V be the total variation of g(x) on the closed interval [ a, /3 ] plus the maximum modulus of g(x) on [ a, /3 ]. Then

fpg(x)e(f(x))dxl5

4V

a

Proof As in Lemma 5.1.2 we need only consider the pure exponential integral. If f'(x) changes sign at a point y in the interval (a, )3), then f'(x) z 8A for x Z y + 8, and I f'(x)I z SA for x < y - S. We use fY+se(f(x))dxl y-s f." < 1/ iTSA + 28 + 1/irSA,

f pe(f(x)) dx 5

8

e(f(x)) dx

+l f p e(f(x))dx y+s

where we have used Lemma 5.1.2 on the first and third integrals. We take

Partial summation for exponential sums

89

S = 1/ (ara) . If y does not exist, or y 5 a + S, or y>- 6 - S, then there is a similar subdivision of the range of integration which leads to a stronger inequality.

Lemma 5.1.4 (rth derivative test) Let f(x) be real and r times differentiable on the open interval (a, f3) with f (')(x) >- µ > 0 on (a, G). Let g(x) be real, and let V be the total variation of g(x) on the closed interval [ a,13 ] plus the maximum modulus of g(x) on [a, R ]. Then

f Rg(x)e(f(x))dxl
- T/C2M2,

so that f'(x) is monotone with A = f'(a) < f'(x) < B = f'(/3). Let g(x) be a real function of bounded variation V on the closed interval [ a, l3 ]. Then g(n)e(f(n)) = a_-5n_-5P

ASrSB

g(xr)e(f(xr) - rxr+ ) g fd(xr)

+O(V+Ig(a)D

77

+log(B-A+2)1

Evaluating exponential integrals

109

where xr is the unique value in a <x 5 /3 with

f'(Xr) = r. Proof First we consider the unweighted sum. We apply Lemma 5.5.2 to the integrals J

Re(f(x) -rx)dx

in the truncated Poisson summation formula (Lemma 5.4.3) with y = Xr. First we consider the exceptional cases when Lemma 5.5.2 gives less information than the Second Derivative Test (Lemma 5.1.3). This occurs when Xr is within a distance O(M/ FT) of the endpoints a or (3. Now -xr)f"()

1 =f'(xr+]) -f'(xr) _ (xr+l for some

between xr and xr+ 1. Hence

(5.5.9) xr+1 -xrXM2/T. The number of values of r with xr within O(M/ FT) of either endpoint is

MT

1, the Riemann zeta function is given by 1

W

(s)= F S. n=1 n

By Lemma 5.4.1 with a = T, b - oo,

(s)

T-1

1

ns

1

T-

1 1

1

+

1

1

+f

2TS

dx+ f

T xs

Tl-S

1

ns + 2TS +

s-1

00 s(s + 1)u(x) X

T

S+z

( s(s + 1)o (x) dx. xs+z + JT

dx (5.5.14)

The integral on the right converges absolutely for Re s > -1. Hence (5.5.14) gives a meromorphic continuation of C(s), and we may use it as the definition of C(s). From the proof of Lemma 5.4.2 we see that

u As + 1)v(x) IT

xs+2

s(s + 1)(1 - cos2.7rrx)

u

dx = fT

2a2r2xs+2

r=1 s

-

1

12

1

US+1 )

( TS+1

E s(s + 1) r=1

1((

21rzrz I TU

/

dx

t

1

xv+zI\eI`rx-

t

+eI -rx- Z-logx T2

1

« Tv + r=1

dx

1 «Tv, r2 Tv+z r 1

1

by the First Derivative Test (Lemma 5.1.2). We let U tend to infinity.

Step two We use the truncated Poisson summation formula of Lemma 5.4.3,

Analytic lemmas

112

with a weight function g(x) =x-° put in as in Lemma 5.5.3. For ease of writing, we consider the case of t positive. We have T

T

1

Fns=E fNe

1

1-

x°dx+O

log T

N°

where the sum is over t

t

1

1 2arT-45r52i7N+4.

The term r = 0 gives T'-g

N'-$

1-s'

1-s

the first term cancels in (5.5.14). The integral with r = [t/27rN + 1/4] is

divided into blocks with x running from Nk to Nk+l, chosen so that

t

Nk+ I < 2Nk for each k. By the Second Derivative Test (Lemma 5.1.3)

fNk"e (

Nk

rx-

2?r

Nk

logx )I

x dx
- T/C2M2. Suppose also that f'(x) changes sign from negative to positive at a point x = y with a < y < P. If T is sufficiently large in terms of the constants Cr, then we have

I Rg(x)e(f(x)) dx -g(a)e(f(y)+8) +g(/3)e(f(/3)) f'"(y) 21rif'(9) +O

T4U11

+-

g(a)e(f(a)) 21rif'(a)

1

N/2(

+O

3) The implied constants are constructed from the constants Cr. 5.6

1

U (1

(T

+

N/2)

BILINEAR AND MEAN SQUARE BOUNDS

An analytic number theorist, it is said, is someone who is very good at using

Cauchy's inequality. Often in analytic number theory we come across a double sum MZ

N2(m)

F,

F,

m=M1 n=Ni(m)

g(m)h(n)e(f(m, n)).

(5.6.1)

Bilinear and mean square bounds

115

Following Montgomery and Vaughan, we classify double sums as type I if either sequence of coefficients is smooth. If h(n) has bounded variation, then we can estimate the inner sum in terms of pure sums. If both coefficient sequences are irregular, then we use Cauchy's inequality. We state the first lemma for a sum slightly more general than (5.6.1). Lemma 5.6.1 (Type II sums) M2

N2(m)

F,

F,

2

g(m)h(m, n)e(f(m, n))

m=Mi n=Nl(m) M

Ig(m)I2) M2 n1

h(m,nl)h(m,n2)e(f(m,n1)-f(m,n2))

E

x

m=M1

n2

N(m)S n U112

Zm)zn1,n2

Proof We use Cauchy's inequality on the sum over m; the second factor is M2

2

N2(m)

E h(m, n)e(f(m, n))

E

m=M1 n=NI(m) rearranged.

In our next lemma a sum in one variable is treated as a double sum. The method is `divide and conquer'; it is equivalent to splitting the range into short intervals, using Cauchy, and then averaging over the possible points of dissection.

Lemma 5.6.2 (Differencing step) Let f(x) be a real function, defined for M< x:5 N, and let w(M), w(M + 1), ... , w(N) be any coefficients. Let D 5 (N - M + 1)/2 be a positive integer. Then N

E w(m)e(f(m))I

(N-M+2D-1)

2 A

N(S) >_

=2

3/2

j

i

f

S

)

f

2 2

2

e(sx - six)sinc2 j

Sx

dx

2 2

S

I.F'(x)I sinc 2 dx Z 2

Sx

C3

IF(X)12

7r/2

dx,

and in the other direction /

z

N(S)5Ei F(2) since1sS 21) ` / / 2

4

26 f E E e(sjx-six)A(2Sx)dx

'72S

0

= 2 C28 11

2S

j

i

f

S

2

IF(x)I A(26x)dx

SIF(x)IZ dx.

p

The k-dimensional case goes similarly.

Lemma 5.6.6 (Double large sieve)

Suppose xWlW,..., x(m) and yU),

are real vectors in k dimensions with

-Xi/2 5 x;"') S Xi12,

-Y/2 < y, ") 5 Y,/2

, y(N)

Bilinear and mean square bounds

121

fori=l,...,k; m=1,...,M; n=1,...,N. Let Ei=1/Xi, Si=X,/(X;Y+1), and put

A(x) = max(1 - Ixl,O). Then 2

E ambne(x(m) Y(")) m

k

4k

17.

(2) A(S)B(e) [1 (X;Y + 1),

(5.6.3)

n

where k

x(m) - x(r) k

A(s) _ E E IamlIarl [l A

r

r

m

L

k

B(E)= '+

bn b

r

n

1

si

1

t

(

AI

y(rn)

E'

I.

y,(r)

Proof As in the last lemma we drop the suffices and treat the onedimensional case. Since E/2

ify- E/2

e(sx) ds = e(xy)

e(Ex/2) - e(- Ex/2)

= Ee(xy)sinc Ex,

21rix

we have

E E ambne(xmyn) E E m

n

m

n

am bn

fYn + E/2

e slnc Exm y,- E/2

= 1(Y+ 0/2

e(sxm) ds

F,

-EAme(sxm)

(Y+ 0/2 m

IYn-sIs /2

where am

Am -

E sine Exm

I AmI S

2E

laml.

By Cauchy's inequality we have ambne(xmyn)

11: m

SI1I2,

n

where I1 and I2 are the integrals

I

z

(Y+E)/2

2

b

I (Y+E)/2I IYn-SISE/2 "

_

bnbrmax(E - IYn n

I1 = f

ds

yrI,O) = EB(E);

r (Y+ E)/2

2

I F, Ame(Sxm)I ds

-(Y+E)/2 M

b,, dS,

Analytic lemmas

122

with 00

7T2

I1 < f 4 since

2

6sl>Ame(sXm)I ds m

00

7r2

rXk -Xm

AmAk4SAI m

k 7T2

IAmAkI m

ir 4 16

k 1

SEZA(S)

46

S

)

maxi`1-

IXk -Xm)

7r4 (XY+ 1) 16

S

,0

A(S).

E

For the k-dimensional case we use Cauchy's inequality inside a k-fold

0

integral with respect to s1, ... , sk.

To show why Lemma 5.6.6 is called the `large sieve', we deduce from it the simplest inequality of sieve theory, the `Brun-Titchmarsh Theorem'. Hardy and Littlewood (1923) found the first result of this type using Brun's selective version of Eratosthenes' method, rejecting multiples of small primes. Later Selberg changed the treatment from linear to quadratic, finding a signed weight function which is large at primes, and summing its square over all the integers in the interval. Linnik meanwhile found sieve upper bounds using his

dispersion method. The basic idea of the dispersion method is to prove uniform distribution by a mean-to-max method, relating the absolute discrepancy to the mean square of the discrepancy on subintervals. Halberstam and Richert (1974) give a detailed account of sieve methods. We give two proofs of the Brun-Titchmarsh theorem from Lemma 5.6.6, the first one of Linnik's type, the second one of Selberg's type. The two proofs are dual in the sense of bilinear forms (see Lemma 10.6.3).

Lemma 5.6.7 (Primes in an interval) There is a constant C such that, out of any M(Z 4) consecutive integers, at most CM/log M of them have no prime factor less than YrM-.

Corollary If the endpoints of the interval are greater than VM_, then at most CM/log M of the integers in the interval are prime.

First Proof Suppose that the vectors y(°) are E-well spaced, in the sense that, if n r, then for some i we have IyI "> -y(r)l > E;.

Then all terms in the sum B(e) with no r are zero, and B(E) _ , Ib"I2. n

(5.6.4)

Bilinear and mean square bounds

123

We choose bn by

k= E ae(x(m) . y(n)). m

The left-hand side of (5.6.3) is B(E)2, so we have k

4k

B(E) _

ame(x(n') .y(n))I2 S (

I

2)

m

n

A(S) F1 (X;Y + 1).

5.6.5)

Now we specialize k to be one (we drop the suffices i), and the points y(n)

to be the fractions of the Farey sequence AQ) with - i < a/q 5 m - (M + 1)/2, so that

X=M-1,

$=1-1/M.

e=1/(M-1),

Y=1,

x(n') _

Since S < 1, we have 12.

A(S) = E la

The minimum distance between members of AQ) for Q >- 2 is 1/Q(Q - 1), so the spacing condition (5.6.4) holds with Q = [vrM-] (the lemma is trivial for

M 4).

Suppose that the interval goes from K + 1 to K + M. We put am =1 if K + m has no prime factor < Q, zero otherwise. When yn = a/q, then

y )= Eame - m-

ame(x

2

=ea(K+M+1))Ea e(a(K+m) q

m

2

m

q

Let

(a)

Eame

q

a(K+m) q

m

which depends only on a/q modulo one. We can now write (5.6.5) as a

Ir 4

2

E E IS(Iq 5-MEIamI2, q=1 amodq 16

(5.6.6)

m

(a,q)= 1

a form of the large sieve familiar to number theorists.

Let R = E am be the number of non-zero terms in S(a/q). Then using Ramanujan's sum (Lemma 1.4.1) we have

S(a)=Eam a mod q

(a,q)=1

q

m

E e\ a mod q

a(K+ m) q

(a,q)=1

= mLam clqE c/ cl(k+m)

(q)= Fla. µ(q)=p p(q)R, m

Analytic lemmas

124

since the only integer c which is a factor both of q and of K + m is one. By Cauchy's inequality (this corresponds to Linnik's dispersion step) we have

F,

p.2(q)R2 5 cp(q)

a mod q

(a,q)=1

Dividing by cp(q) and substituting in (5.6.6), we have 2

R

µ2(q)

ir4MR

cP(q)

16

F,

q--,Q

15 .6.7)

The last step is to estimate the sum in (5.6.7) from below. The Mobius function µ(q) is 0 unless q is a product of distinct primes, Pi, ... , p, say, and then

µ2(q)

1

1

r

cp(q)

cp(q)

Pi...P,

1

1

(1

Pu

1

)

r( II 1+-+ zR +... _ -,n Pi PI Pr 1 1

1

1

1

taken over all integers n whose prime factors are all taken from the set (Pi,. .. , p,). Hence

µ2(q) q ,Q

1

97(q) -

L

n

taken over all integers n for which the product of the distinct prime factors of

n is an integer < Q. Thus µ2(q) q

Q (q)

Q

1

nz

Q+1

dx x

= log(Q + 1)

iz log M.

(5.6.8)

We deduce the result with C = ir4/8. Second proof Similarly, if the vectors x(m) are 6-well spaced, then 4k

2

bne(x('n) y("))1 "I

n

5

k

(2) B(e) F1 (X;Y + 1).

We take k = 1, x(m) and y(") as in the first proof, and choose b by

µ(q) a M+ 1 9P(q)e(q(K+ 2 ))

(5.6.9)

Bilinear and mean square bounds

125

where y(") is the fraction a/q. The points y(") are e-well spaced, so that

B(e) _

µ2(q) _

':

Q µ2(q) P(q)

q=1 amodq PT(q) (a,q)=1

Thus (5.6.9) gives 2

µ(q)

E F, m q=1 (P(q) amodq

e

7r 2M Q µ2(q)

r a(K+m)) q

5

1

(a,q)=1

16

E 1

cp(q)

(5.6.10)

which is Selberg's sieve inequality (with a worse constant). The sum over a is the Ramanujan sum again. In particular, if K + m has no prime factor 5 Q, then the sum over a is µ(q). The left-hand side of (5.6.10) is µ2(q)

> R 1

2

q'(q)

and we complete the argument using (5.6.8) as in the first proof.

The form of the large sieve in Lemma 5.6.6 is due to Bombieri and Iwaniec (1986a). The application in Lemma 5.6.7 follows Mathews (1973). Lemma 20.2.2, which bounds the number of times that an exponential sum is large, is also a sieve result in the widest sense. The other lemmas were either known

to van der Corput, or are later developments of his ideas. For example, Lemma 5.5.2, from Huxley (1994c), corresponds to the leading terms only of van der Corput's asymptotic formula (1935, 1936), without taking f(x) to be infinitely differentiable. There are many forms of Lemma 5.5.2, with different hypotheses about the function f(x), for example, Lemma 10 of Heath-Brown

(1983), Lemma 2 of Karatsuba (1987), and Lemma 3.4 of Graham and Kolesnik (1991). The more specialized Lemmas 10.6.1, 10.7.1 and 21.4.3 are also of van der Corput type.

6 Mean value results 6.1 THE SIMPLE EXPONENTIAL SUM The idea in this chapter is to investigate a sum S by introducing a real parameter x, and then evaluating E2 = f 1IS(x)I2 dx. 0

Apart from its own interest, the integral has three uses. 1. Limitation results. Since IS(x)I >_ E for some value of x, any uniform upper

bound for IS(x)I must be greater than E. These results are called omega

theorems from the notation F = O(G) for `F does not have order of magnitude strictly smaller than G'. 2. Upper bounds. Sometimes there is a mean-to-max argument, using special properties of S(x), as in the Tauberian theorems of convergence theory. 3. Large values results. If x1, ... , xR is a well-spaced set of points at which IS(x,)I is large, then R itself cannot be too large. The large sieve (Lemma 5.6.6) is often used in this way. Theorem 6.1.1 Let f(x) be a real function with R continuous derivatives for M 5 x 5 2M, for which there is a size parameter T and a constant C >_ 1 such that If (')(x)I 5 CrT/M' (6.1.1) and

if W(x)I z T/CSMS

(6.1.2)

hold for certain values of r and s. Then f2 I

ZM1

2

dt=M+O(CM2

L e(tf(m))

Tg M

M

provided that R Z 1 and (6.1.2) holds with s = 1, and 1 f1

2M-1

2

E e(f(m +x)) dx=M+O

C2M2 log M C9M4 T

+

T2

,

M

provided that R >_ 3 and (6.1.1), (6.1.2) hold with r = 2,3 and s = 2, and f (3)(X)

The simple exponential sum

127

does not change sign. The term O(M4/T 2) in the second result may be improved if (6.1.1) holds for larger values of r.

Proof We have

f

2

12Mf 1

2

f2

dt =

e(t f (m)) M

e(t(f (m) - f (n))) dt m

M+

n 1

m

O n

f(m)f(n)I

m#n

by the First Derivative Test (Lemma 5.1.2). Since

f (m) - f (n) = (m - n)f'(e ) for some

between m and n, the error term must be
IkI, then we treat y as the independent variable, and argue similarly. We split the range at Mat and M)31, where u =Mat corresponds to the tangent angle +yo it/3, and u = M/31 corresponds to the tangent angle 1 = 7r/3. On the range Mf31 to M/3 we use various forms of the first derivative test. Since we have F" = 1/p cos3 41,

F' = tan 41,

3 sin 4-

F(3)

pcos54

dp

1

p2cos341

dcr'

integrating by parts introduces improper integrals as 41 tends to it/2. We split the range of integration at u = M/32, corresponding to 41 = 4/2 with

cos +lr2 =1/ (kM) . Then, by the First Derivative Test (Lemma 5.1.2),

fMp e(kMF(t/M) + ht) Mpg

21rik

dt
M,

f1DM(u, v) du dv >

provided that R z 4. The implied constants depend on the curve C.

Proof We drop the suffix M. By Parseval's formula for Fourier series we deduce that

f111 0

0

(N(u, v) -AM2)2 du dv =

F, F, Ia(h,k)12 h

k

(h,k)#(0,0)

(R -1)2NT.

(7.2.6)

As we shall see, R gives the usual order of magnitude of the denominator q when we approximate the real number f" (x)/2 by a rational number a/q. From (7.2.4) and (7.2.6) we have R2 >- N. (7.2.7) The Bombieri-Iwaniec method improves the classical bounds for exponential sums when R N. (7.2.8) We assume that M and T are sufficiently large (in terms of the constants Cr)

for R and N to exist and be large themselves. It is convenient to express orders of magnitude in terms of M, N, and R instead of M, N, and T. For

(1-SW Sx5(2+8)M we have N4f (4)(x) - 1 be a real number. In the notation of Lemma 7.4.2, we have Y)1)2

E(1) e(-x(h) 1

a/q

ABH5 NR2V(1
> 1 for i = 1, 2, 3, whilst X4Y4 = 2r (2 H)

QN

VQ27

We ought to treat the range Q >> N, the flanks of the major arcs, separately, taking out the term in h1/2 in (7.4.1) by partial summation (Lemma 5.2.1),

and applying the large sieve to three-dimensional vectors. However, the largest contribution comes from ranges with Q x R. We have 4

F1 (X;Y+1)

«l1+

Q1

f 4

1

The simple exponential sum

158

Lemma 5.6.6 allows us two coefficient sequences. We take the coefficients on the vectors x(h) to be 1 in case 1, and 1 or 0 in cases 0 and 2, according to whether the vector x(h) occurs in the sum E('). We take the coefficients on the y vectors to have unit modulus, and argument opposite to that of the sum

r(l) e(-x(h) .y), h

so that we get the sum of moduli. We have

5i 2Zr and

M'

N3 Zr

< < 17T

1

1

X

(7.6.9)

.v . I

I µgN3I

3g µg3zrl

We have

r- K

-h(N3)=(qzr-N3)h"(6)^µgN3zr-

q

N

q

for some 6 between N3 and qzr. Hence (7.6.9) implies

r - K - qh'(N3) e(hg(n)) H N,

H3 N3

where

H)I

HN

HN

C«I1+HII+HN)+HN«1. II

13

llllllway

Taking moduli in this than

means that we cannot expect any estimate better

M

N

HM (HN)
) form a coincident pair if their corresponding entries differ by numerically less than 1

1

1

FK

8KLV' 4L (2K) ' 4L ' 8L The result is still true for a family of sums Si of the form (8.1.1), taken over subintervals (depending on i) of H< h < 2H, and of M < m < 2M, formed with functions fi(x) satisfying (8.1.3) uniformly in i, when we replace b(j) by Ni, j), y(J) by y('°>), and the sum over j by a sum over i and j. The sequence of rational numbers aj/qj corresponding to the minor arcs may also depend on i.

proof This is Lemma 5.6.6 with

X= (8KLV, 4L (2K) , 8L,16L/V), and 1

Y= 1,2max

(µq3)

1

,1,max

.

(l-+-q3)

we can take Y = Y3 = 1 since the rational numbers a/q and ab/q are only defined modulo one. By (8.1.3) and (8.1.4) we have

2) 3)
-1/C, IF(')(x)I 5 Cr, (9.1.2) x 5 2, the upper bound holding for r = 2, 3, 4, 5, and the lower bound

Exponential sums with a difference

186

for r = 2, 3 always, and for r = 1 if M>> FT. We write d1 for d/dx and f1 for d1 f, etc., so that (9.1.2) gives Idi-1f(x,Y)I< C Mr

Id,-1f(x,Y)I < 2MT

r

for M+ 2H - 1 <x < 2M - 2H and the corresponding values of r. We suppose that M is sufficiently large, and that integers N and R can be chosen with 15 R :!g H:!' N < M to satisfy 2R2NT >-

C3M3

> 2(R -1)2NT,

(9.1.3)

2C3{H + 1 2K - 1, and has a pole of order 2 or 3 at s = 2K - 1. The pole has order 2 for forms of even weight and congruence level one.

Corollary

When the pole has order 2, then there is a constant B2 with L

1=1

I b(l)4

lzk- 1 5 BZ (log2L)z

for any positive integer L.

Either Lemma 10.3.2 or 10.3.3 can be used to improve the smoothing error in Lemma 10.3.1.

10.4 WILTON SUMMATION ON THE FAREY ARCS We apply the truncated Wilton summation formula (Lemma 10.2.6), and evaluate the resulting exponential integrals.

Wilton summation on the Farey arcs

211

Lemma 10.4.1 (Explicit Wilton summation) We have (when f" (x) > 0) M2

E wj(m)b(m)e(f(m)) j=1 m=M

_ E 1F, F, b(l)el - 8111 wj(xj + O BMk/2

I MZRZ

N3

)hil

M

(x, )e(gj± (xj ))

11/(2s-2)

+ RZ

)(logM)12),

)

(10.4.1)

where xji and x,-.I are the stationary phase points satisfying

f'(x)=a± 1 (I), qj qj V x

(10.4.2)

the functions g j W, g,-., ( x) are given by

ax

2

qj

qj

(IX)

g (x)=f(x)- '--+-

+8+8,

the weight functions hit (x), h7 (x) are given by

(x/l

)ck-1)/2

(f" (x) ± (l/x3)1/2/2qj)

(4lxgf )1/4

and the length of the reflected sum K1 is the smallest integer with

k1+1 _ 18M/R2. Proof First we fix j, and write a/q for aj/qj. We apply the truncated Wilton summation formula with a(m), b(l) replaced by

b(m)e(am/q), q-kb(l)e(-al/q) as in Lemma 10.2.3. The size of the cut-off K depends on the derivatives ax

dxr

wj(x)e f(x) - q

which are sums of terms involving r,

q

with

ro+r1 +2r2+

+tr1=r.

We have w,(ro)(x) , and

uj =

f'(u12)=ai/q1,

x+-o,

and A0,..., A4 are constants constructed from the constants C1.

Proof Using Lemma 5.6.1, we bound the square on the left of (10.5.10) by 2L-1 lb(l)12

lkh,(l)ht(1)e(gi(l) -ge(l))

1k

1

t

The first factor is O(B2 log L) by Lemma 10.2.1. We estimate the sum over 1 in the second factor using Lemmas 5.1.1 and 5.4.3. The weight function

H(l) = lkhi(l)h1(l) is allowed to depend on i and j. We need estimates for the size and total

variation of H(l) that are uniform in i and j. The bound for dxj,/dl in Lemma 10.5.1 shows that the weight function H(1) is f times a slowly varying function of 1, with

H(1)^

Mk 1NR2 Q

l M1 ,

and the total variation of H(l) has the same order of magnitude. After Lemma 5.1.1, we consider sums e(g,(1) - g,(1))

(10.5.15)

A large sieve on the Farey arcs

217

over subintervals of the range L < I < 2L. We have d ge(l)

dl

_ - gjV(x1l) -

-

and 1

xj

2qj

(13)

d2

dl2

ge(l)

qj

)

dx

lxj, 1

dl

1

1

2qj

,

By Lemma 10.5.1 we have kdl

a, 1 uj gi(1>+- -+-+

d

qj f

dl

32C2 NR

+q

71

2

(10.5.16)

15MQ2

A5N2R4

1

2ql

q1

aj

uj

1

ge(l) + q,

a uif, (ui)

2

MaQs

0; O(NE) solutions;

x log N solutions.

In practice, the nature of the problem seems to change as soon as we add an inequality. The algebraic geometry becomes less relevant. It may be that most solutions of the inequality satisfy not just the given inequality, but a

corresponding equation, and they lie on an algebraic variety of smaller dimension. Many divide-and-conquer arguments go from an inequality to an

equation in this way. For the equations (11.1.1) - (11.1.4), if a and 0 are small enough, then we expect the diagonal solutions to predominate; these are the solutions with g1,. .. , g, equal to hl,..., h, in some order. We start with two variables. Let d(m) denote the divisor function as in Section 2.1 of Chapter 2, the number of ways of writing m as a product of two positive integers. Lemma 11.1.1 (Binary quadratic forms) Let n be a positive integer. Then the number of pairs of integers x, y (of either sign) with x2 - y2 = n is 2d(n) if n is odd, 2d(n/4) if n is a multiple of four, and zero if n is twice an odd integer. The number of pairs of integers x, y with x2 - xy +y2 =n is

6 E X(d), din

where X(d) is the Dirichlet character mod 3 taking the values X(d) = 0, ± 1 according to X(d) = d (mod 3). Corollary If a and b are integers with a2 < 3b, then the number of triples of integers hl, h2, h3 (of either sign) with

hl+h2+h3=a, h; +h2 +h3 =b is

(d

(6-3X(a2))

XI deven,

(11.1.6)

dl(3b-a2)

Proof If x2 - y2 = n, then x + y = e, x - y = d for some integers d and e of the same sign, and either d or e both even, or d and e both odd. Conversely, each such pair of d and e gives a solution. When n is odd, then any pair of integers with de = n gives a solution. When n is even, then if de = n, either d or e must be even. If n is twice an odd number, then the other factor must be

Preparation: divisor functions

237

odd, and we cannot construct a solution. If n is a multiple of four, then any pair of even integers d and e with de = n gives a solution. The equation x2 - xy + y2 = n is treated in the same way in principle, but the factorization is x + py = 6, x + p2y = e, where p is a complex cube root

of one, with p2 =1- p, and 6 and a are algebraic integers of the form a + bp, where a and b are integers (see Hardy and Wright 1960, Chapter 12).

The lemma contains the theorem that every prime number of the form 6m + 1 can be written as x2 - xy +y2. For the corollary we eliminate h3 to get

2h; + 2h2 + 2h1h2 - 2ah1- 2ah2 = b - a2, r and we note that the substitution

x=h1+2h2-a,

y=h2-h1

(11.1.7)

makes

x2 - xy +y2 = 3hi + 3h2 + 3h1h2 - 3ah1 - 3ah2 + a2 = (3b - a2)/2. (11.1.8)

Conversely, if x and y satisfy (11.1.8), and if a is a multiple of three, then (x +y)2 is a multiple of three, and so is x +y, and we get integer values for h1 and h2 from (11.1.7). If a is not a multiple of three, then

2(x +y)2 - -a2 = 2a2(mod3). Half the solutions have x + y = a, and the other half have x + y = -a. Only the solutions with x + y = - a give integer values for h and h2 in (11.1.7). 1

We deduce the formula (11.1.6).

In the next few lemmas we see that the sum over d in Lemma 11.1.1 is not too large, especially if we can average over n. Thus when h4, .... h 91, ... , gr have been chosen in (11.1.1) and (11.1.3), then the last three variables take

care of themselves. We need the more general divisor function d,(n), the number of ways of writing the positive integer n as a product of r positive integers. We give the standard bounds. The averages in Lemma 11.1.3 may be replaced by asymptotic formulae. The leading terms in the asymptotic formu-

lae may be obtained by elementary arguments as in Lemma 11.1.3, or by counting lattice points as in Chapter 2, or by using the Dirichlet series generating function

d,(n) ns

Lemma 11.1.2 (Bounds for divisor functions) We have

d,(mn) 5 d,(m)d,(n),

(11.1.9)

with equality if the highest common factor (m, n) is one, and

d,(n)ds(n) S d,s(n).

(11.1.10)

The ruled surface method

238

For any S > 0, there is a constant B(r, 8) with

dr(n) SB(r, 6)NS.

(11.1.11)

Proof If m = k1k2 ... kr,

and n = 1112 ... lr, then mn = g1g2 ... qr, with q1= k,!1. If (m, n) = 1, then q, determines k. _ (q,, m) and 1, = (q,, n). If (m, n) > 1, then there are several sets of factors k1,.. . , kr and 11, ... , Ir which give the same products q1, ... , qr. This proves (11.1.9). If n = k1k2 ... kr = 1112 ... ls, then we construct rs numbers q1j whose product is n as follows. For any factor e of n, we put ml(e) = (k1, e), and, for i > 2, m1(e) = (k1k2 ... k1, e)/(k1k2 ... k;-1, e). Then q,1 = m,(11), and, for j > 2, q,1= m1(1112 ...11)/m,(1112 ... 11_ 1).

The fraction q,1 has the form

(kk',Ii')(k,1) (k, ll')(kk',1)

with k=klk2

k1_1, k' =k1, 1=1112

1i_1, 1' =1,. Let d=(k,l), k=du,

1= dv. Then (u, v) =1 and

(uk', vl') (uk', vl') q11= (u, vl')(uk', v) (u, l')(k', v) ' which is an integer. It is easy to check that the rs numbers q,, determine

kl,...,kr and When n is expressed as a product of powers of distinct primes

n=pSI...pkk, then

dr(n) = fl dr(p,'). For p a prime, dr(ps) is the same as the number of ways of typing a sentence

of s words on paper with r lines. This is the number of ways of typing r + s - 1 things, each either a word or a carriage return, with s words and r - 1 carriage returns. Thus d"(ps) _

(r+s - 1! Y1H--, we have r )=1

r

($t - h))=

rj (x2 - y?) = O(?1Hr). )=1

Interchanging the roles of x; and y) gives the corresponding inequality with C] g, and h) interchanged.

We can now deal with the case r = 4. This account is based on Watt (1989a). We give the simpler argument which loses a factor H. The proof has a divide-and-conquer structure. We divide the integer solutions into families, and count the number of families which have many solutions. This brings in another important idea, the Riesz interchange, which is the discrete analogue of the transformation

f13If(x)Idx= f/3 a

suplfxI flf(x)Idydx= fy=o

x=a y=o

f

dx dy,

where S(y) is the set of values of x for which I f (x)I ? y. This idea is useful when we can describe the set S(y). Theorem 11.2.5 The number of solutions of (11.1.1), (11.1.2), and (11.1.3) with r = 4 in the range (11.1.5) is

O(H4 + 6Hs+E) for any e > 0. The implied constant depends on e.

Proof By Lemma 11.2.3, the solutions fall into families. There are O(H4) solutions belonging to the families that contain at most 768 solutions.

Families of solutions

247

Families with more than 768 solutions are divided into blocks according into which range L to 2L - 1 the number of solutions falls, where L = 21 is some power of 2. We note that there are at most 24H4 trivial solutions in which 91, ... 194 form a permutation of h1, ... , h4. If (after renumbering) we have g4 = h4 and g3 = h3, then the solution must be trivial, since

gi

+g2 hi + h2, =

g1 + 92 = h1 + h2,

and so g1g2 = h1h2, and

(h1 -g1)(h1 -g2) = h; - (h, +h2)h1 0. Then either h1= g1 (whence h2 =g2), or h1=g2 (whence h2 =g1). If one solution in a family is trivial, then all solutions are trivial. Hence we can refer to trivial or non-trivial families. We estimate the number of non-trivial families with at least L solutions of

(11.1.3). We can normalize the family (renumbering if necessary) so that 0,g,z0,h4=0. By Lemma 11.2.4 with r = 4, 91929394

(H-r)r ways, and then have g1, ... , gr a permutation of h1, ... , hr in r! ways; these are diagonal solutions. For the second bound, we divide four-dimensional space into boxes B,,, and we let n; be the number of vectors x(h) (in the notation of Lemma 7.4.2) in

each box. Two vectors (distinct or not) falling into the same box give a solution of (11.1.1)-(11.1.5). Let I be the number of boxes. Then Cauchy's inequality gives (Ln,)z

0, with constant depending on e. Proof Equation (11.1.1) implies (11.3.1) with r= 1, and (11.1.1) and (11.1.3) together imply (11.3.1) with r = 2. Since 5

(x - a1)...(a - a5) =x5 + E (-1)rxs-r(a1,..., a5), r= 1

then we have

I(x-h1)...(x-h5) - (x-g1)...(x-g5)I S7SH5 for HS x < 2H. In particular 1(g; -hI)...(g; - hs)I-_ 1,

We put

f3=(b+K)/q,

We suppose that U >> H, since the case U > 1/H4 or Ig(4)(x)I >> 1/H5 at each point of J. Suppose that J has length L. In both cases we have Ig(3)(x)I >>L/H5, I g" (x)I >>L2/H5 somewhere on J. The length of J is O( (o-K1Y5) ),

which tends to zero as promised. We estimate trivially on all intervals whose length is O(H1/2U1/6). This leaves O(log H) long intervals to consider. We apply Poisson summation (Lemma 5.4.6) to get ah2 + bh e

+g(h)

q

G(a, b - r; q) qA-rSgB+-',

f e(g(x) - rx) dx !

q

log(gB-qA+2)), with

qB+qA 1,

(12.3.26)

then N10(6, A) 0 is arbitrary, but the implied constants depend on e.

Proof We fit Lemmas 12.1.2, 12.2.2, and 12.3.2 together to estimate the integral in (12.1.2). We have

0 H'

1 N10(2U«

U(Ql/4 +

U4/3

Hs+e(1+

0

N

For

H

k

+

k

V`-

1

2 k; +

J

k; + kj

1

2k

(k;kj)

2

so that d

2dt

1

logo

1

(klk4) + (

(k1k3)

k2 - k1)( k3 + (k1k2k3k4)

1

1

(k2k4)

(k2k3)

k4 ) >> IM1-M21 z

K

Hence within a fixed family, the values of t for which (13.1.3) holds form an interval of length ,K4 O I m1(m1 - m2)(m1- m3)I

which gives the bound (13.1.12). Similarly considering the values of 14/12 leads to the bound (13.1.11).

0

The First Spacing Problem for the double sum

276

Lemma 13.1.4 (Solutions of all four equations) Suppose that k1,. .. , k4, ll, ... ,14 satisfy (13.1.1)-(13.1.5). Define 0 and .p by

kl +12 k2 -13 k3 -14 k4 = OLFK,

ll

11/ kl + 12/ k2 - l3/ k3 -14/ k4 = cpLFK (so that 0 and cp are 0(17)). Then

max rjIki-kjl P, I ql >- Q.

O

14.2 MAGIC MATRICES One of Bombieri and Iwaniec's deepest insights was that the first condition (14.1.1) was linear in the action of the modular group on primitive integer vectors, described in Section 1.3 of the first chapter. To simplify the notation, we write r/q = l a/ql. Lemma 14.2.1 (The magic matrix) in their lowest teens with r

Suppose that r/q and r'/q' are rationals r' 0, then since

neither k/m nor (l - k)/(n - m) lies in J, but I/n, which is in J, lies between these two rationals, then we have

1-k

k

2SA<m -- n-m

1

m(n-m)'

so that SAmn < 1;

this inequality holds trivially if m = 0. We rewrite (14.2.6) as r

r'

1

9 , q, _ - + O(6A). n

(14.2.12)

The First and Second Conditions

294

Then (14.2.2) and (14.2.3) give

an - c1= n9+ O(SAnK) = O(SAnK), do + c1= n/O + O(SAnK) = O(SAnK). Hence by (14.2.11)

m

ce

ce

n

n

n

am - ck= -(an-cl)- - =m6- - +O(SAKm), and similarly ce

ck+dm =m/9+ - +O(SAKm). n

The magic matrix is conjugate to

d')

(c'

n)-1(c

=(k

l n1

(m

d)

akn + bmn - ckl - dim - akm - dm2 + ck2 + dkm

aln + bn2 - c12 - din

-aim - bmn + ckl + dkn

1

(an - cl)(ck + dm) - mn

c

-(am-ck)(ck+dm)+m2

(an-cl)(cl+dn)-n2 -(am-ck)(cl+dm)+mn

Here mn

a'

c

SAnK

+O

Icl
Q H1H12-H11H2

10_3

(F111)

A family of sums

305

In the construction of Chapter 10 for P5 Q M F112 H H12 H11H2 4T2 x3(F11)" 1

and in the case P > Q 1

4M

H1H12 -H11H2

D3 x3(F11)3

Lemma 14.4.3 (Coincidences from a family of sums) Suppose that the family of sums is formed with functions F(x, y,) which satisfy (14.4.1) and the conditions of Lemmas 14.3.1, 14.3.5, and 14.4.1. Then the number of coincidences in the First and Second Conditions among minor arc vectors from the whole family

of sums withP> 1 and A2 PQ >> 1 by Lemma 14.1.1.

In Lemma 10.6.5 we need the maximum number of coincidences involving a particular minor arc. Lemma 14.4.4 (Coincidences with a fixed minor arc) Under the hypotheses of Lemma 14.4.3, the number of coincidences involving a fixed minor arc with

P> N/M (15.2.8) by the condition N3 rQ'

t

We can rewrite (15.3.11) as

a'u

6'

v

t + R, - t 5 Q' for some integer v, where 6' «D4. We have set up the conditions of Theorem 4.3.3 with parameters a', (3',

6', M', N', and R'. Either R' ::g max(86'Q',1285'M'Q'2) + 1, (15.3.13) which is impossible if G(u/t) is less than the first term in the minimum in (15.3.1) of the lemma with B3 sufficiently large, or else all the solutions lie on a rational line ey = cx + d, with c, d, and e integers with no common factor, and l

ea

86'Q' rG « N2, - c sR'-1

(15 . 3 . 14)

l

and

1e13-dl
> 1 is (16.2.5)

SPQ >> k.

Lemma 16.2.1 (Type 1 matrices acting on congruence families) For a congruence family of sums of the form (16.2.3) or (16.2.4), satisfying (16.2.5), the number of coincidences between pairs of minor arc vectors indexed by rational

numbers a/gwith PS al > 1. Choosing Q = 256r + 1 gives r >> N, contradicting the assumption r < NIB, if the constant B1 has been chosen sufficiently large. Hence the points do lie

on a straight line, which we can write as Ix + my + n = 0 with integer coefficients satisfying m - 5 different minor arcs, then we replace (R2 + rs)2 by R4/L2.

Proof For an upper triangular matrix (

I

B), the Second Condition in the

form (16.3.12) gives (aq)

h "(a)

=3

µ

-

1

µ

2 3

(f l3)(g(a/q +B)) 4B

=-3

f (3)(g(a/q))

f (4)(g(g)) (f(3)(g(, )))3

for some q. Upper triangular matrices occur only for M < FT, when we have assumed that IF(4)(x)I is non-zero, so that /

h"(allxA=IBIM2R 4

(16.3.14) q

The Third and Fourth Conditions

346

Lower triangular matrices

('

° ) only occur for M >- FT, when we have

assumed that 3(F(3))2 - F" F(4) is non-zero, and a similar calculation gives

xA=ICIM.

(16.3.15)

We have set up the conditions of Lemma 4.2.1, with h(x) in place of f(x),O(A4) in place of 3, and the distance A between the rational points a;/q; satisfying

A >> min(1/R2,1/rs). We have Ik(1) +nl > L/R2, and in (4.1.14) of Lemma 4.1.5 we can take both factors to be of size >> L/R2, so that

L2/R4 «04/i Q, and we have

A4/Q «A4R4/OLZQ2, and we still omit the term 04/Q. As in Chapter 15, we shall apply these lemmas with either e/r or f/s as the rational number of smallest denominator that is a value of f" (x)/2 on the minor arc. In the second case, e/r is the rational number of second

smallest denominator, and (e - f)/(r - s) is outside the minor arc, but adjacent to f/s in some Farey sequence. In both cases we find that rs 5 3R2.

The construction of Chapter 8 gives corresponding results, but with different derivatives and factorials in (16.3.1) and (16.3.3), and, for example, O5 _ 1 for M>> Ts/1z The conditions (17.1.11) ensure that (17.1.18), (17.1.19), and the conditions of Chapter 7 can be satisfied by a suitable choice of N and R in the ranges (17.1.10). For M > T'/ 12, we take V = 1, and choose N and R so that 1 x O1P2 x M2/N4, which puts N and R into the ranges (17.1.16). The conditions (17.1.17) ensure that (17.1.18), (17.1.19) and the conditions of Chapter 7 can be satisfied by a suitable choice of N and R in the ranges (17.1.16).

We can now substitute the estimates for A and B into Lemma 7.4.3, then the result into Lemma 7.4.2, and add up the cases. A complication is that the bound for A5 changes when Q x N1- E, and that we must change the bound for B from Lemma 16.1.3, valid for Q Q,, then a valid upper bound is given by the order of magnitude for Q x R, multiplied by R'".

Proof In Lemma 7.2.2, the largest power of Q which occurs in W(Q) is Q-1. Hence in Lemma 7.4.2 there is a factor q-3r/2-1+s outside the expression from the large sieve. When Modification 2 operates, then we can insert a

factor R2/Q2, since the sum over a/q contains >> Q2/R2 approximating polynomials corresponding to each minor arc. In the expression ABH5NR2V 3

Q

(17.1.24)

The simple exponential sum

353

of Lemma 7.4.3, we can ignore the factor 1 + Q/N for small Q. Since H NQ/R2, and the largest exponent of H in the bound for A is 2r - 3 + E, then the largest exponent of Q in (17.1.24) is 2r - 1 +a+ e for Q > N. We can now work out the largest exponent of Q in Lemma 7.4.2 as a + E

2

-$-

r + 7

2

for Q > N. Hence a positive choice of 6 is possible under the conditions given. If the upper bound for ABV has a discontinuity at Q0Yby a factor Qo, then we define an exponent 0 by (Qo/R)B ^ Q'7,

and insert an extra factor (Q/R)B into (17.1.24). We can still choose a if (17.1.23) holds: otherwise we use the larger estimate with the factor Q'I over the whole range Q >> R.

We can now give bounds for the simple exponential sum S of Chapter 7. Further complications arise when we consider a family of sums Si. Theorem 17.1.4 Let F(x) be a function four times continuously differentiable on the interval 1 <x :!g 2, whose derivatives satisfy the following conditions:

I F(')(x)I 5 Cl

(17.1.25)

I F(')(x)I >_ 1/C1

(17.1.26)

for r = 3, 4,

for r = 3, where Cl is a positive constant. Let 6:5 1 be a positive real number, and let M2

e(TF(mM)),

S= ME

where M and M2 are positive integers, T is a large real number, and M <M2 < (1 + 5)M < 2M < T. Suppose that for some C2 >- 1 either case 1 or case 2 holds:

Case 1 M 5

and (17.1.26) holds for r = 4 also;

Case 2 M >- C21 FT and (17.1.25) and (17.1.26) holds for r = 2 also and IF" (x)F(4)(x) for some positive constant C3.

-

3(F(')(X))21

>- C3

(17.1.27)

Exponential sum theorems

354

If M is sufficiently large in terms of C1, then we have S _ 2, C5 (which may depend on S), a and /3 such that, for a class C of functions closed under the addition of linear terms and under linear changes of variable, we have in the notation of Theorem 17.1.4 ISI C4T ", we have

ISI52C5MTa-a.

Exponential sums with a parameter

357

Proof For M5 T" we pick an integer q > 1 so that

gMST"

J

1/57

(T8

>> f 133

M2 T + (Mz + 7 Jlog M,

(17.2.5)

1

We also have, for M > M112T 1/6 >> M2/3.

Hence DR Z M113, and the estimate for the original sum using R differencing steps must be >> Ml- 1/3X28+'

For T large, we take R large, so we are far from the root mean square size

V.

Lemma 17.4.1 (Smoothing the differences) Let D, L, Q, and R be positive integers. Let A be a set of integers lying between L and 2L - 1. Suppose that for each 1 in A we are given R positive integers d1(l),... , dR(l) with

d1(l)d2(l) ... dR(l) =1, d1(l) + +dR(l) 0, then the conditions of Theorems 17.1.4 and 17.4.2 hold for each R. We write M = T a. The bounds take the form S _ 1). The classical notation states bounds as S _ 1 when (18.1.6) holds. When (18.1.6) is false, then we choose V= 1, which puts N and R into the ranges (18.1.9), satisfying (18.1.21), and also satisfying (18.1.18) in the range (18.1.11). With V chosen so that A1P2 x 1, the last term in (16.1.10) of Lemma 16.1.3

dominates the first term if 1 « L)2 24 3Q/P X Q213R4/H513M.

(18.1.22)

The extremal choice of N and R is (18.1.13), and V>_ I when (18.1.12) holds.

Exponential sums

375

When (18.1.12) is false, then we choose V= 1, which puts N and R into ranges (18.1.15), satisfying (18.1.22), and also (18.1.18) in the range (18.1.17). We have to satisfy (8.1.4)-(8.1.7), so that

R :s- H5 N/64C,

(18.1.23)

R >- 2C H _+1

(18.1.24)

,

N<M/10.

(18.1.25)

The requirement (18.1.23) gives the conditions (18.1.4), (18.1.8), (18.1.10), (18.1.14), and (18.1.16). We find that, up to a constant factor, (18.1.24) follows from (18.1.23) and the condition A1Q2 >> 1 which our construction always satisfies. The constant B2 is chosen large enough for (18.1.24) to hold with

the right constant of proportionality. Also (18.1.25) follows easily from (18.1.18) if M and T are sufficiently large. The most difficult verification is (18.1.18) for the choice (18.1.7), where we use both lower bounds (18.1.6) and (18.1.10) in different ranges. Similarly, both (18.1.12) and (18.1.14) are used to verify (18.1.18) for the range (18.1.13). 0 Theorem 18.1.2 Let F(x) be a real function three times continuously differen-

tiable for 1 <x5 2, and let g(x), G(x) be bounded functions of bounded variation on 1 5 x S 2. Let C0,. .. , C6 be real numbers z 1. Let H, M, and T be large parameters. Suppose that

IF(')(x)I < C,

(18.1.26)

forr= 1,2,3, IF(')(x)I

1/C,

(18.1.27)

for r = 1, 2, and that either Case 1 or Case 2 holds:

Case l M5 C0T 1I2 and (18.1.27) holds for r = 3 also, Case 2 M >- Co 'T 1/2 and IF'F(3) - 3F" 21 Z 1/C4.

Let S denote the sum

S= E g( H

H)

m=M

G(M)e(M

F(M)).

Then there are positive constants B1 and B2, depending on C0,. .. , C6, and on the functions g(x) and G(x), such that for M in the range CS 1T49/114 SM5 C5T651114

and H satisfying

H C-1 T 1/2. Let N(S) denote the number of solutions of T

(18.2.1)

m IIM F(M )IIS8

with M:5 m 5 2M - 1. Then in both cases/ (18.2.2)

N(S) 2, followed by a rigid motion. Then, for any embedding of E in the Euclidean plane, the number of integer points in E is M)3151146),

AM2 + O(IM46/73(log where I is a number depending on the curve C, but not on Moron the position or orientation of E.

Theorem 18.3.3 In Theorem 18.3.2, suppose that the pieces C, are four times differentiable, in the sense that p is twice continuously differentiable with respect

to the tangent angle 41. Suppose also that p is either constant or has a finite number of maxima and minima on each piece Ci, and that d log p/d 4' has a bounded number of maxima and minima between any two consecutive extreme values of p on Ci. Then we may take

I= E r' pk/73' i

k

Lattice points inside a closed curve

389

where the inner sum is over local maxima Pik (including endpoints, and counting constant values only once) of the radius of curvature on Ci, provided that M is so large that the bounds M > 1 / p and M40

63 Z 1+

(log M)"

1

P

2

dP

2 103

d/,)

1 1

hold on each curve C1.

Proof of Theorems 18.3.2 and 18.3.3 We have to choose local coordinates on each piece Ci so that the conditions of Theorems 18.2.1 and 18.2.2 apply. The new coordinates are u and v in Lemma 18.3.1, with

u = r (x cos /3 +y sin f3 ),

v=W(ycos/3-xsin so, for (x, y) on the curve MCi, we have dv

- = tan(4,- /3),

(18.3.7)

du

d2v

due d3v

du3

1

r1/epcos3(f-/3) 3sin(4r-/3) 1 dp rp2cos5(ap-p) rp3cos4(q-/3) dq, 3sin(4i- f3-A) rp2

ip - /3 )cos A '

(18.3.8)

(18.3.9)

where the angle A is defined by

dp 3p d+/ 1

tan A =

We also have

dv d3v - 3 (d2v 2 du du3 due

3cos(4,-/3 -A) rp2 cos5(r/l - /3 )cos A '

(18.3.10)

We prove Theorem 18.3.2 by a convexity argument. The values of 4 on a

particular piece C. of the curve form a closed interval K. For any 4 in K there is an open interval of values of f3 for which the expressions in (18.3.8) and (18.3.9) are bounded away from zero and infinity. We pick a(41) to be the midpoint of this interval. For fixed 4 = 4i0, we take f3 = a(410) and

consider (18.3.8) and (18.3.9) for this P. Since p and A are continuous functions of +/, there is an interval of values of 41 about 1//0, open relative to the interval K, on which neither expression (18.3.8) nor (18.3.9) is zero or infinite. We call this interval I( /3 ). Each 41 inK lies in its own open interval

Lattice points and area

390

I(a(t/i)). There is a finite subcollection I1,...,IR of these intervals which covers K; we suppose that the subcollection is minimal, and numbered in order, so that I, nIr+2 is empty, but I, fl Ir+ is non-empty. We form closed intervals Jr by taking the midpoint of I, fl I,+ as the common endpoint of J, and Jr+ 1. Then J1,..., JR are closed intervals, disjoint except for endpoints,

that cover K, each corresponding to some angle /3 = ar. Let Sr be the smallest distance from an endpoint of Jr to the corresponding endpoint of Ir. The derivatives in (18.3.8) and (18.3.9) are finite and non-zero for 0rEJr+[-6r/4,6r/41. 1 /3-arl 0, uniformly for MN)")1"(K+1)

N< T < MN(MANK(log

(18.5.7)

Then M2

R=

.

M

m

p(NF( M))
_ N >_ N4, and for N8 >_ N >_ Nlo if either of these ranges is non-empty, M(62Q-llq-33)/(62Q-2)N11/(62Q-2)(log N)6(5g2+6)/(31Q-1)

R
0, C > 0, and so t > 0. We see from Lemma 14.3.2 that t is bounded. We use Lemma 16.3.5 with rs «R2. If a coincidence extends over L z 1 consecutive minor arcs, then Lemma 13.4.2 allows us to replace A2 by O(A2/L2) in Lemma 14.3.2, so that t > FT and V chosen so that 1 > FT V.

Corollary A bound of the form (19.3.2) valid for a certain class of functions and a certain size range for (log N')/(log M') implies two other bounds of the same type, valid for related classes of curves and certain size ranges. In the first bound, the exponents (Ki, A,) are (1/13, 9/13), (1/5, 2/5), and two terms

2+3K+2A 14+23K+24A (22+33K+32A' 22+33K+32A)'

12-2K (23+3K+A' 22+3K+A 3+3K+A

corresponding to each term M'AN" in (19.3.2). In the second bound, the exponents (K;, A.) are (1/5, 7/15), (0, 3/4), and two teens

4+8K+9A 12+18K+17A 24+38K+39A'24+38K+39A)' corresponding to each term in (19.3.2).

1-K 13+3K+A

(11-K' 22-2K

Further results

420

Lemma 19.3.3 (Iteration step for the double sum) The choices of N in Lemma 19.3.1 give bounds for the sum S of Chapter 8:

S > fT .

Corollary A bound of the form (19.3.2) valid for a certain class of functions and a certain size range for (log N')/(log M') implies another bound of the same type, valid for related classes of curves and certain size ranges. The exponents (Ki, A.) are (1/6, 1/2), (1/2, 0), and two terms

1+2K+A 1+5K+6A (4+9K+9A' 4+9K+9A)'

3+4K+A

3-3K

(12-K-A' 12-K- A)

corresponding to each term M'AN" in (19.3.2).

The proofs of Lemmas 19.3.2 and 19.3.3 follow those of Theorems 17.1.4

and 18.1.2. The corollary to Lemma 19.3.2 uses Lemma 18.5.2, and the corollary to Lemma 19.3.3 follows the proof of Theorem 18.2.1. The two cases

of Lemma 19.3.3 give the same exponents in (19.3.2) with M' and N' interchanged.

The new exponents from the iteration are not very sensitive to the values of K and A, and are useful only for a certain range of (log N')/(log M'). In practice this range is a short interval of the longer range in which the bounds are valid. We start the iteration from the bounds of Table 18.1, except that we use Theorem 18.2.1 in place of Theorem 18.2.2 and Lemma 18.5.2 for the

range 83/635 a5 643/466. Moreover, we can use the analogue of Theorem 18.2.1 for a family of exponential sums, using t as the parameter. This is the only range in which we can use t as a parameter. The bounds lower down

Exponential sums with a large second derivative

421

Table 19.1 Bounds for integer points close to a curve Iterated

Steps

D DD DDDC

23

23

41

73

73

32 =

17

35

7

3

41

396348

79

79

17

17

32

309023

DCC

DC

69787

3118

6577

396348

49115058

146195

146195

14735

14735

309023

38213083 -

15603

40525

3143

8083

49115058

1590221

84191

84191

16837

16837

38213083

1229946 =

1251

3306

115

342

1590221

3799577

6829

6829

679

679

1229946

2859937 =

3118

6577

598

333

3799577

8366056

14735

14735

1589

1589

2859337

6185661 =

3143

8083

675

1571

8366056

269897

16837

16837

3363

3363

6185661

197842

342

4

43

269897

27

679

679

64

64

197842

19 =

598

333

29

62

27

1375774

1589

1589

140

140

19

939695 =

675

1571

4

43

1375774

44151

3365

3365

64

64

939695

29740 =

4

43

44151

19

64

64

29740

12

29

62

19

374

140

140

12

227

89

356

374

872

641

641

227

423 =

C

1.2812... 1 . 2826

...

1 .2853

...

1 .2929

...

1.32 88...

1

.3525

...

=1.3573.. .

115

CC

C

=

27548

DDCC

DDC

from

Exponents K, A

Range for (log N)/(log M) from to

1

.

4211 ...

1

.

4641 ...

1 . 4846

...

=1.5833...

=

1

.

6476 ...

2 0615 .

...

Table 18.1 already use Theorem 17.2.2, with a parameter from the differencing step or from the Fourier series for the row-of-teeth function. Tables 19.1 and 19.2 give some further bounds for integer points close to a curve and for exponential sums. In the first column D means the double sum, C the simple sum, and A the differencing step, so that DCC means Lemma 19.3.3 applied to the result of Lemma 19.3.2 applied to one of the bounds that uses Theorem 17.2.2, labelled C(1) in Table 18.1. As usual, rows bracketed together correspond to an upper bound with two or more terms. Estimates for exponential sums in other ranges can be obtained by the A (differencing)

Further results

422

Bounds for exponential sums

Table 19.2

Iterated

Steps

Exponent (3

C

from

Range for a = (log M)/(log T) from to

89 + 285a

106822

139817

570

246639

246639

= 0.5669...

2387 + 17972 a

115

342

1033325

106822

27290

679

679

2642746

246639 =

2819 + 19177«

598

CDC

0.4331...

333

699371

1033325

2642746 =

29855

1589

1589

1647930

11897 + 88442 a

675

1571

156527

699371

134680

3363

3363

370694

1647930

CCC

= 0.4244...

113 + 897a

4

43

263

156527

1345

64

64

638

370694

CC

CC

0.4288...

= 0.4222...

491 + 3624«

29

62

143

263

5530

140

140

349

638 -

569 + 1053a

29

62

307

143

2800

140

140

641

349

1273 + 2484«

89

356

68682

307

6410

641

641

171139

761

AC 1

0.4122...

= 0.4097.. . = 0.4034.. .

4 + 103a

12

68682

128

31

171139 =

29 + 173a

227

12

280

601

31 =

0.4011...

0.3871.. .

or B (reflection) steps as in Table 17.1. We get some improvement near the values of a for which the steps in Table 17.1 change from A'C to A'+ 1C. For most values of a, the saving obtained from the iteration is less than the saving from using Theorem 17.2.2 instead of Theorem 17.1.4 after a sequence

of differencing steps. The iteration works best for exponential sums with a = (log M)/(log T) near 5/12, where the third derivative is still small, but the second derivative has order of magnitude larger than one. This range of a provides the title for this section.

20 Sums with modular form coefficients 20.1

EXPONENTIAL SUMS

We obtain the main results of Jutila (1987a et seq.) for the sum M2

S = E b(m)e(f (m))

(20.1.1)

M

of Chapter 10, where E b(m)e(mz) is a cusp form of even weight k for the full modular group. Lemma 20.1.1 (Choosing parameter sizes) Suppose that the conditions of Lemma 14.3.1 hold. Then, in the notation of Lemma 10.5.1, we have R 13/2

B(01(V ), 02(V ), 03(V ), 04(V )) > MR2, and z

P

04(V) X VT

M LQ2

+ PQ

( M) >> i,3(V).

To avoid double suffices, we write yi for the value of the parameter on the Farey arc with index i. We have U.

-=1+O(03), ui and

f11(u3,Yi)

= 1 +O(D4).

f1i(uj ,Yy)

Since F12 and F112 are supposed bounded and non-zero, f11(u?,Yi)

f11(ui3, Yj)

1 XIYi -y,IX

f1(ui,Yi) z fl(u; ,Y;)

-1

Sums with modular form coefficients

428

Now

f1(ul,Y1) =fl(u),Yj) +b, so that bQ

i,Yi) P - fl(u),yj) .f1(uz

-1 «lyi-Yjl+L3

_ 1/C1

for r = 1, 2, where C1 is a positive constant. Let G(x) be four times continuously

differentiable for all x, and zero outside the range 1 5x 5 2. Let b(m) be the coefficients of a cusp form of even weight k for the modular group, and let 2M-1

/

M=M

`

S(t) = m b(m)GI M)el tF( )), `

/

where M is a positive integer. Let C2 be a positive constant, and let T be a real number with M5 C2T. Then f2TIS(t)I6

dt V. For each power of 2, V, we have

I(V)
N/M or N'/M). Then S(t) is larger by a factor VM_ than the sum we want, so when (19.1.9) of Theorem 19.1.3 holds, then the squared integral for each block is O(ff). Blocks with M small are estimated as in Theorem 18.2.3. There are O(log T) blocks, so Cauchy's inequality gives O(A log2 T) for the whole integral.

Much deeper arguments give the corresponding mean value theorem T

Jo

+(2y-1)T+E(T),

with an explicit estimate for the error term E(T). We use the machinery of Heath-Brown (1979), following Heath-Brown and Huxley (1990).

Lemma 21.3.2 (Truncated mean square expansion) Suppose that 15 A S T1/3. Let t - 21rm2

E

g(t) = 2

ms (t/2a)

(m/n)"

G(t) = 2 m

n

m 1

(mn) ilogm/n

exp

02 log2 m/n 4

the conditions of summation in G(t) being

0 < I log m/ni 5 (log T)/0, mn -K>0 on[-/3,/3], and for 25k5r+2 If (k)(x)I < /31-kIf'(x)I. Then

1

e(f(x))cosr

R R

-x

dx
1, p prime, A(m) = 0 if m is not a prime or a prime power. The basic property of these weights is E A(d) = log m. (22.1.4) dim

We use the prime number theorem in a weak form (see Davenport 1967 and Hardy and Wright 1960 for the two standard proofs). Lemma 22.1.1 (Prime number theorem) For P > 2 we have

E A(m) =P+O msP

P log P

Let I be a subinterval of 1 <x < 2, chosen so that no combination of derivatives of F(x) occurring in the proof vanishes on I. We work with the integers in the interval MI, the set of points of I multiplied by M; let the endpoints of MI be M1 and M2. Let E(2) denote a sum over integers m for which (22.1.2) has a solution. Then (2)

E A(m) _ MI

A(m)(f(m + 1) -f(m)) MI

+ L A(m)(p(-f(m + 1)) - p(-f(m))).

(22.1.5)

MI

By the prime number theorem and partial summation, the first term in (22.1.5) becomes

f(M2)-f(M1)+0 logM

(22.1.6)

.

On the left of (22.1.5) (2)

(2)

E A(m)

F, log p + OI

MI

r

A(m)(22.1.7) /

m52M,mnotprime

P,-Ml

in this chapter p is a variable of summation restricted to prime numbers. The second sum in (22.1.7) is

E E log P< p

rz2

log 2 M

log p

log P

p'S 2M

>

e(-hf(m)) - e(-hf(m + 1))

f(m+1)

2,irih

f(m)

e(-ht) dt

=e(-hf(m))

ff(m+1)-f(m)e(-ht)dt. 0

There is some constant c with

f(m + 1) -f(m) > Q, so that p >> M112. This can be achieved by interchanging q and r within each block of the sum if necessary. With the combinatorics of Section 22.1 we take z close to M112, so that w is bounded, to achieve the same effect. For p >> M 1/2, the necessary condition (22.3.10) holds for M
. Then (23.5.5) implies

a-2 a; ( A

1

6

m

H44

111-

za

+0 0+ M J

(23.5.7)

)

Fouvry and Iwaniec study (23.5.7) by exponential sums. We can also use Farey sequence methods. For fixed A, the fraction m2/m1 lies in a short

interval. We take e/r to be the rational number of least height in this interval, and let f/s, f'/s' be its neighbours in the Farey sequence .9(r). Now m2/m1 lies in one of two reference intervals. For m2/m1 between e/r

and f/s, we have m1= ru + st, m2 = eu + ft for some integers t, u as in Lemma 1.2.2, but the common factor (t, u) = (m1, m2)'may be greater than one. We can also write

A = (e +f9)/(r+s9), where 0 may be negative. Substituting in (23.5.7), we find that (t - Ouxru + st) lies in an interval whose length is proportional to A. This is an integer point

close to a curve problem. The trivial estimate O((6 + 1XL + 1)) (in the notation of Chapter 3) gives

O(OM2 +H4/M2 +M/r), and by Lemma 1.2.3, we have r _ 2 of Lemma 24.2.2 are elementary, but the case c = 1 depends on counting the number of points on an algebraic curve over finite fields.

The terms of the Kloosterman sum correspond to matrices

of the modular group, and Kloosterman sums occur in the Fourier coefficients of modular forms. Kuznietsov (1980) followed ideas of Selberg to express sums of Kloosterman sums explicitly in terms of the Fourier coefficients of modular forms.

Further ideas

482

Lemma 24.2.3 (Kuznietsov's trace formula) For a, b positive integers, g(x) a continuous piecewise differentiable function of compact support q 2

(ab))

g(2

(ab)

a )kk-1/2

1)k

°°

+k=1 E

21T

b (-)

K(a, b; q) + Sab f J.

x

g(x) dx x

2

(cb(G2k a) - Sab)(2k -1) f J2k-1

4 (ab) bt(a)bt(b)g(K,)

c

1

+

cosh in.1

j=1

(2)

x)

g(x) dx

T;t(a)Tj,(b)g(t)

?r J- (1 + 2it)( 1 - 2it)

dt,

where G2k,a is a modular form of weight 2k, the ath Poincare series, and c is its nth Fourier coefficient (suitably normalized), b,(n) is the nth Fourier coefficient of the jth non-holomorphic Maass wave form (suitably normalized), and a + Kj is the corresponding eigenvalue, and Ta(n) is a normalized divisor function

F, E

T.(n)

d

()a,

e

de=n

Sab is 1 for a = b, 0 otherwise, and (x2

1T i

g(t)= 2sinhirt f (J2it(2)

where JS(y) is the Bessel function. In the same notation q 2 vV-Wab)

-

g

2

K(-a, b; q)

(ab)

4 (ab) b,(a)b,(b)g(Kt) j=1

cosh?rK,

T,t(a)T;t(b)g(t)

1

+ 1rL

dt,

with

g(t)= 2 cosh1rt f Kh2it(x)g(x)dx, where

Khf(y) _

2

W

fo exp(-y cosh t)cosh st dt

IT

1

=

1

I

exp(-2(' +u))u-s

du

is a less familiar Bessel function, the Hankel function of pure imaginary argument (Jeffreys and Jeffreys 1962, cap. 21).

Subdivision without absolute values

483

The two transforms g(t) and g(t) are analogous, since we can express Khs(y) in terms of 13(y) - I_S(y). The Kloosterman sum is a discrete analogue of the definition of Bessel functions of integral order as integrals round the unit circle. We call Lemma 24.2.3 a trace formula, as the right-hand side can be regarded as the normalized trace of an intertwining operator in representation theory. The lemma is proved using the Fourier theory of functions on the hyperbolic plane invariant under the action of the modular group PSL(2, 7L)

as rigid motions, and the Poincare series coefficients enter as residues at trivial poles.

Bombieri and Iwaniec (1986a) suggest that a full solution of the Second Spacing Problem will involve Lemma 24.2.3. We have calculated directly with

the matrices rather than with their Fourier theory. There are three integers a, b, and q in the Coincidence Conditions of Part IV. Perhaps 3 X 3 matrices will be needed. The representation theory of SL(3, 71) is still being investigated.

Are we on the right road to showing that the discrepancy of a smooth closed curve of linear dimensions M is O(MI/z+E)? Subdividing the curve according to rational gradients must be right. Perhaps the number of Farey

arcs should be ME, and we take about 1/e terms of the Taylor series. Alternatively, the y vectors may be so uniformly distributed that the contri-

butions of different minor arcs cancel. The Kuznietsov trace formula is appropriate here because SL(2, 7L) is the automorphism group of the lattice

points, a natural part of the problem. Should the representation theory of SL(n,7L) for n >_ 3 appear in a two-dimensional problem? In all events, we have reached the final formula: `If you want any more, you must sing it yourself.

References The numbers flush right following a reference are the chapter or section numbers in which the reference is cited, or to which it is related. Atkin, A. O. L. and Lehner, J. (1970). Hecke operators on 170(m). Math. Annalen, 185, 134-60. 10,20 Atkinson, F. V. (1949). The mean value of the Riemann zeta function. Acta Math., 81, 10, 21.5 353-76. Baker, R. C. (1986). Diophantine inequalities. Oxford University Press. 5

Baker, R. C. and Harman, G. (1991). On the distribution of apk modulo one. Mathematica, 38, 170-84.

23.3

Baker, R. C., Harman, G., and Rivat, J. (1994). The Piatecki-Shapiro Theorem. J. Number Theory, 50, 261-77. 22 Balasubramian, R. (1978). An improvement of a theorem of Titchmarsh on the mean square of I C(1/2 + it)I Proc. London Math. Soc., 36, 540-76. 21.4 Bombieri, E. (1987). Le grand crible dans la theorie analytique des nombres, 2nd edn. Asterisque, Paris. 5.6

Bombieri, E. and Iwaniec, H. (1986a). On the order of C(1/2 + it). Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4),13,449-72. 5,7,8,14,16,17,21,24.2 Bombieri, E. and Iwaniec, H. (1986b). Some mean value theorems for exponential sums. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13, 473-86.

7.5,12

Bombieri, E. and Pila, J. (1989). The number of integral points on arcs and ovals. Duke Math. J., 59, 337-57.

23.1

Branton, M. and Sargos, P. (1994). Points entiers au viosinage d'une courbe plane a tres faible courbure. Bull. des Sciences Maths. (2) 118, 15-28. 3, 23.1 Cassels, J. W. S. (1953). A new inequality with application on the theory of Diophantine approximation. Math. Annalen, 126, 108-18. 5,6 Cassels, J. W. S. (1959). An introduction to the geometry of numbers. Springer, Berlin. 11.2

Chaix, H. (1972). Demonstration elementaire d'un theoreme de van der Corput. Comptes RenduesAcad. Sci. Paris, A 275, 883-5.

2

Chandrasekharan, K. and Narasimhan, R. (1967). On lattice-points on a random sphere. Bull. Amer. Math. Soc., 73, 68-71. 6 Colin de Verdiere, Y. (1977). Nombre de points entiers dans une famille homothetique de domaines de R. Ann. Sci. Ec. Norm. Sup. (4)10, 559-76. 6 Corput, J. G. van der (1920). Uber Gitterpunkte in der Ebene. Math. Annalen, 81, 1-20. 2,5 Corput, J. G. van der (1921). Zahlentheoretische Abschatzungen. Math. Annalen, 84, 53-79. 5 Corput, J. G. van der (1922). Verscharfung der Abschatzung beim Teilerproblem. Math. Annalen, 87, 39-65. 3,5 Corput, J. G. van der (1923). Neue zahlentheoretische Abschatzungen. Math. Annalen, 89, 215-54.

5

References

485

Corput, J. G. van der (1935). Zur Methode der stationaren Phase I. Compositio Math., 1, 15-38. 5.5 Corput, J. G. van der (1936). Zur Methode der stationaren Phase II. Compositio Math., 3, 328-72. 5.5 Corput, J. G. van der and Pisot, C. (1939). Sur la discrepance modulo un I. Proc. Kon. Ned. Akad. Wetensch., 42, 476-85. 5.3, 5.6 Davenport, H. (1967). Multiplicative number theory. Markham, Chicago. 5.4,21,22.1 Deligne, P. (1974). La conjecture de Well. Inst. Hautes Etudes Sci. Publ. Math., 53, 273-307. 10.3, 20.1 Deshouillers, J. M. (1976). Nombres premiers de la forme [n`]. Comptes RenduesAcad. Sci. Paris, A 282, 131-3. 22.1 Deshouillers, J. M. (1985). Geometric aspect of Weyl sums. Elementary and Analytic Theory of Numbers. Banach Centre Pub. 17 (Polish Sci. Pub; Warsaw) 75-82. 2 Deschouillers, J. M. and Iwaniec, H. (1982). Kloosterman sums and Fourier coefficients of cusp forms. Inventiones Math., 70, 219-88. 24.2 Everest, G. R. (1989). A `Hardy-Littlewood' approach to the S-unit equation. Compositio Math., 70, 101-18. 23.3

Faltings, G. (1983). Endlichkeitssatze filr abelsche Varietaten fiber Zahlkorpern. Inventiones Math., 73, 349-66.

11.1

Filaseta, M. (1988). An elementary approach to short interval results for k-free numbers. J. Number Theory, 30, 208-25. 3.2 Filaseta, M. and Trifonov, O. (1996). The distribution of fractional parts with applications to gap results in number theory. Proc. London Math. Soc. To appear. 23.1 Ford, L. R. (1938). Fractions. Amer. Math. Monthly, 45, 586-601. 1.2 Fort, C. (1973). New lands. Ace, New York. 15.2

Fouvry, E. and Iwaniec, H. (1989). Exponential sums for monomials. J. Number Theory, 33, 311-33.

23.5

Gel'fond, A. O. and Linnik, Yu. V. (1965). Elementary methods in analytic number theory. Rand McNally, USA.

6.3

Good, A. (1982). The square mean of Dirichlet series associated with cusp forms. Mathematika, 29, 278-95. 20.3 Good, A. (1983). Local analysis of Selbeig's trace formula. Lecture Notes in Mathematics, 1040. Springer, Berlin. 21.1 Graham, S. W. and Kolesnik, G. (1991). Van der Corput's method for exponential sums. London Math. Soc. Lecture Notes, 126. Cambridge University Press. 3,5,7,14,17.4,18.2,19.2,23.4

Hall, R. R. and Tenenbaum, G. (1986). The average orders of Hooley's 0r functions, II. Compositio Math., 60, 163-86. Hall, R. R. (1988). Divisors. Cambridge University Press.

11.1,18.5 11.1

Halberstam, H. and Richert, H.-E. (1974). Sieve methods. Academic Press, London. 5.6

Hardy, G. H. (1915). On the expression of a number as the sum of two squares. QuarterlyJ. Math. (Oxford), 46, 263-83.

10.2

Hardy, G. H. and Landau, E. (1924). The lattice points of a circle. Proc. Royal Soc., A 105, 244-58.

10.2

Hardy, G. H. and Littlewood, J. E. (1923). Some problems of `partitio numerorum'; III: on the expression of a number as a sum of primes. Acta Math., 44, 1-70. 5.6,12.1

Hardy, G. H. and Ramanujan, S. (1918). Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. (2) 17, 75-115. 7.1,12.1

486

References

Hardy, G. H. and Wright, E. M. (1960). An introduction to the theory of numbers, 4th 1,11, 22.1 edn. Oxford University Press. Heath-Brown, D. R. (1978). The twelfth power moment of the Riemann zeta function. 21.5

Quarterly J. Math. (Oxford), 29, 443-62.

Heath-Brown, D. R. (1979). The fourth power moment of the Riemann zeta function. 21.3 Proc. London Math. Soc., (3) 38, 385-422. Heath-Brown, D. R. (1983). The Pjateckii-Sapiro prime number theorem. J. Number Theory, 16, 242-66.

5.5, 22

Heath-Brown, D. R. and Huxley, M. N. (1990). Exponential sums with a difference. 9,19.1, 21.3 Proc. London Math. Soc. (3) 61, 227-50. Herz, C. S. (1962a). Fourier transforms related to convex sets. Annals of Maths., 75, 6.2, 23.4 81-92. Herz, C. S. (1962b). On the number of lattice points in a convex set. Amer. J. Math., 6.2, 23.4 84,126-32. Hlawka, E. (1950). Integrale auf konvexen Korpern. Monatshefte Math., 54, 1-36, 6.2,18.5, 23.4 81-99. Hlawka, E., Schoissengeier, J., and Taschner, R. J. (1991). Geometric and analytic number theory. Springer, Berlin. Hobson, E. W. (1957). Theory of functions of a real variable. Dover, New York.

23 6.3

Huxley, M. N. (1972). On the difference between consecutive primes. Inventiones 20.2 Math., 15, 164-70. Huxley, M. N. (1975). Large values of Dirichlet polynomials III. Acta Arithmetica, 26, 435-44. 20.2 Huxley, M. N. (1985). Introduction to Kloostermania, Elementary and Analytic Theory of Numbers. Banach Centre Pub. 17, (Polish Sci. Pub., Warsaw), 217-306. 24.2 Huxley, M. N. (1987). The area within a curve. Proc. Indian Acad. Sci. (Math. Sci.), 97, 2 111-16. Huxley, M. N. (1988). The fractional parts of a smooth sequence. Mathematika, 35, 292-6. 3 Huxley, M. N. (1989a). Exponential sums and the Rieman zeta function, in Theorie des Nombres, C.R. C.I. TN. Laval 1987. de Gruyter, Berlin, 417-23. 3,17 Huxley, M. N. (1989b). The integer points close to a curve. Mathematika, 36, 198-215. 3

Huxley, M. N. (1990). Exponential sums and lattice points. Proc. London. Math. Soc. (3), 60, 471-502. Corrigenda, ibid. (3) 66, 70. 8,18 Huxley, M. N. (1991). Exponential sums and rounding error. J. London. Math. Soc. (2), 43, 367-84. 17.4,18

Huxley, M. N. (1992a). A note on short exponential sums, in Proc. Amalfi Conf. Analytic Number Theory. University Press, Salerno, 217-29. 14,17 Huxley, M. N. (1992b). Integer points in a domain with smooth boundary. in Seminaire de theorie des nombres Paris 1989-90. Birkauser, Basel, 93-111. 18.3

Huxley, M. N. (1993a). Exponential sums and the Riemann zeta function IV. Proc. London Math. Soc. (3), 66, 1-40.

4,15,17, 21.2

Huxley, M. N. (1993b). Exponential sums and lattice points II. Proc. London Math. Soc., (3), 66, 279-301. Corrigenda, ibid. (3), 68, 264.

18

Huxley, M. N. (1994a). The rational points close to a curve. Ann. Scuola Norm. Sup. Pisa Cl. Sci, (4) 21, 357-75. 4 Huxley, M. N. (1994b). A note on exponential sums with a difference. Bull. London Math. Soc., 26, 325-7. 13.3,19.1, 21.3

References

487

Huxley, M. N. (1994c). On stationary phase integrals. Glasgow Math. J., 35, 354-62. 5.5

Huxley, M. N. (1995a). A mean value theorem for exponential sums. J. Austral. Math. Soc., A, 59, 304-7. 6.1 Huxley, M. N. (1995b). The mean lattice point discrepancy. Proc. Edinburgh Math. Soc., 38, 523-31. 6

Huxley, M. N. and Kolesnik, G. (1991). Exponential sums and the Riemann zeta function III. Proc. London Math. Soc. (3), 62, 449-68. Corrigenda, ibid. (3), 66, 302. 11,12

Huxley, M. N. and Kolesnik, G. (1996). Exponential sums with a large second derivative. To appear. 16.3,19.3 Huxley, M. N. and Sargos, P. (1995). Points entiers au voisinage d'une courbe plane de class C". ActaArithmetica, 59, 359-66. 23.1 Huxley, M. N. and Trifonov, O. (1996). The square-full numbers in an interval. Proc. Cam. Philos. Soc., 119,201-8. 3, 23.1 Huxley, M. N. and Watt, N. (1988). Exponential sums and the Riemann zeta function. Proc. London Math. Soc. (3), 57, 1-24. 1.6, 7,14,17 Huxley, M. N. and Watt, N. (1989a). The Hardy-Littlewood method for exponential sums. Coll. Math. Soc. Jdnos Bolyai, 51, Number Theory. Budapest 1987. North Holland, Amsterdam, 173-91. 7,14,17

Huxley, M. N. and Watt, N. (1989b). Exponential sums with a parameter. Proc. London Math. Soc. (3),59,233-52. 14.4,17 Huxley, M. N. and Watt, N. (1994). The number of ideals in a quadratic field. Proc. 18, 24.1 Indian Acad. Sci. (Math. Sci.), 104, 157-65. Ireland, K. and Rosen, M. (1982). A classical introduction to modem number theory. 10.3 Springer, Berlin.

Ivi6, A. (1985). The Riemann zeta function. Wiley, New York.

21

Iwaniec, H. (1980). Fourier coefficients of cusp forms and the Riemann zeta function. 21.5 Seminaire de Theorie des Nombres (Bordeaux) 1979/80. Iwaniec, H. and Mozzochi, C. J. (1988). On the divisor and circle problems. J. Number Theory, 29, 60-93. 5, 8,13,18 Jarnik, V. (1925). Uber die Gitterpunkte auf konvexen Kurven. Math. Zeitschrift, 24, 500-18. 2,3

Jeffreys, H. and Jeffreys B. S. (1962). Methods of mathematical physics, 3rd edn. Cambridge University Press. 7.3,8.3,10.2,21.2,24.2 Jutila, M. (1985). On exponential sums involving the divisor function. J. reine angew. 10, 20.3 Math., 355, 173-90.

Jutila, M. (1986). On the approximate functional equation for C '(s) and other Dirichlet series. Quarterly J. Math. (Oxford) (2), 37, 193-209.

5.5,10, 20.3

Jutila, M. (1987a). Lectures on a Method in the Theory of Exponential Sums. Tata Institute Lectures in Maths. and Physics 80. Springer, Bombay.

10,20

Jutila, M. (1987b). On exponential sums involving the Ramanujan function. Proc. Indian Acad. Sci. (Math. Sci.), 97, 157-66.

10,20

Jutila, M. (1989). Mean value estimates for exponential sums, in C. R Joumees Arithmetiques Ulm 1987. Lecture Notes in Mathematics 1380. Springer, Berlin, 120-36.

10,17.4, 20

Jutila, M. (1990a). The fourth power moment of the Riemann zeta function over a short interval. Coll. Math. Soc. Jdnos Bolyai, 51, Number Theory. Budapest 1987. 10,13.3,14.4, 21 North Holland, Amsterdam, 221-44.

488

References

Jutila, M. (1990b). Mean value estimates for exponential sums II. Arch. Math., 55, 267-74. 10,20 Jutila, M. (1990c). Exponential sums related to quadratic forms, in Proc. Canadian Number Theory Conference Banff 1988. de Gruyter, Berlin.

10,20

Jutila, M. (1991). Mean value estimates for exponential sums with applications to 10,20,21 L-functions. Acta Arith., 57, 93-114. Jutila, M. (1992). Transformations of exponential sums, in Proc. Amalfi Conf. Analytic 10, 24.2 Number Theory. University Press, Salerno, 263-70. Karatsuba, A. A. (1981). On the distance between adjacent zeros of the Riemann zeta function lying on the critical line. Proc. Steklov Inst. Math., 3, 51-66, from Trudy 7.6,21.4 Mat. Inst. Steklov, 157. Karatsuba, A. A. (1987). Approximation of exponential sums by shorter ones. Proc. 5.5 Indian Acad. Sci. (Math.-Sci.), 97, 167-78. Kendall, D. G. (1948). On the number of lattice points inside a random oval. Quarterly 6 J. Math. (Oxford), 19,1-26. Kendall, D. G. and Rankin, R. A. (1953). On the number of points of a given lattice in 6.2,23.4 a random hypersphere. Quarterly J. Math. (Oxford) (2), 4, 178-89. Korobov, N. M. (1992). Exponential sums and their applications. Kluwer, Dordrecht. 5,23 3,5,23.4 Kratzel, E. (1988). Lattice points. Deutscher Verlag Wiss., Berlin. Kratzel, E. and Nowak, W. G. (1991). Lattice points in large convex bodies. Monatsh. 23.4 Math., 112, 61-72. Kratzel, E. and Nowak, W. G. (1992). Lattice points in large convex bodies H. Acta 23.4 Arith., 62, 285-95.

Kuznietsov, N. V. (1980). Peterson's hypothesis for cusp forms of weight zero and Linnik's hypothesis; sums of Kloosterman sums. Mat. Sbomik, 111, 334-83. 14.2,21.1,24.2 Landau, E. (1915). Uber die Gitterpunke. in einem Kreise (II). Gottinger Nachrichten 10.2 (1915), 161-71.

Lax, P. D. and Phillips, R. S. (1967). Scattering theory for automorphic functions. Academic Press, London. Linnik, Yu. V. (1941). The large sieve. DokladyA.N.S.S.S.R, 30, 292-4.

21.1 5.6,22.1

Littlewood, J. E. (1912). Quelques consequences de l'hypothese que la fonction i(s) de Riemann n'a pas des zeros dans le demi-plan R(s) > 1/2. Comptes RenduesAcad. 21.1 Sci. Paris, 154, 263-6. Littlewood, J. E. (1986) Littlewood's miscellany (ed. B. Bollobas). Cambridge Univer7.4 sity Press.

Liu, H.-Q. (1993). On the number of Abelian groups of given order (supplement). Acta Arith., 64, 287-96.

23.5

Liu, H.-Q. and Rivat, J. (1992). On the Pjateckii-Sapiro prime number theorem. Bull. 22, 23.5 London Math. Soc., 24,143-17. Logan, I. and O'Hara, F. (1983). The complete Spectrum ROM disassembly. Melbourne 18.5 House, Richmond, Surrey. Matthews, K. R. (1973). Bilinear forms and the large and small sieves. J. Number 5.6 Theory, 5, 16-23. Meurman, T. (1988). On exponential sums involving the Fourier coefficients of Maass 10,20 wave forms. J. reine angew. Math., 384, 192-207.

Meurman, T. (1990). On the order of the Maass L-functions on the critical line. Coll Math. Soc Janos Bolyai, 51, Number Theory. Budapest 1987. North Holland, 10,20 Amsterdam. 325-54.

References

489

Meurman, T. (1992). A simple proof of Voronoi's identity. Asterisque, 209, 265-74. 10 Montgomery, H. L. (1971). Topics in multiplicative number theory. Lecture Notes in Mathematics. 227. Springer, Berlin. 5,21 Montgomery, H. L. and Vaughan, R. C. (1974). Hilbert's inequality. J. London Math. Soc. (2), 8, 73-82. 5

Moreno, C. J. and Shahidi, F. (1983). The fourth moment of the Ramanujan Tfunction. Math. Annalen, 266, 233-9. 10.3, 20.1 Muller, W. and Nowak, W. G. (1991). Lattice points in planar domains: applications of Huxley's discrete Hardy-Littlewood method, in Number-Theoretic Analysis II. Springer Lecture Notes in Mathematics, 1452, pp. 139-64. Springer, Berlin. 23

Nowak, W. G. (1985a). On the average order of the lattice rest of a convex planar domain. Math. Proc. Cam. Philos. Soc., 98, 1-4. 6.3 Nowak, W. G. (1985b). An 11-estimate for the lattice rest of a convex planar domain. Proc. Royal Soc. Edinburgh, 100A, 295-99. 6.3 Ogg, A. (1969). Modular forms and Dirichlet series. Benjamin, New York.

10

Piatecki-Shapiro, I. I. (1953). On the distribution of prime numbers in sequences of the form [f(n)]. Mat. Sbomik, 33, 559-66.

22

Phillips, E. (1933). The zeta function of Riemann: further developments of van der Corput's method. Quarterly J. Math. Oxford, 4, 209-25.

5

Rankin, R. A. (1939). Contributions to the theory of Ramanujan's function T(n) and

similar arithmetical functions II: the order of Fourier coefficients of integral modular forms. Proc. Cam. Phil. Soc., 35, 357-72. 10.2, 20.1 Rankin, R. A. (1977). Modular forms and functions. Cambridge University Press. 1.3,10 Sargos, P. (1995). Points entiers an voisinage d'une courbe, sommes trigonometriques courtes et pairs d'exposants. Proc. London Math. Soc., (3) 70, 285-312. 3,19.3 Selberg, A. (1942). On the zeros of Riemann's zeta function on the critical line. Arch. for Math. og Naturvid., B 45, 101-14 (Collected Works 1, 142-55). 21.4 Selberg, A. (1956). Harmonic analysis and discontinuous groups in weakly symmetric

Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc., 20, 47-87 (Collected Works I, 423-63). 21.1,24.2 Sierpinski, W. (1906). Sur un probleme du calcul des fonctions asymptotiques. Prace Mat.-Fiz., 17,77-118. 2 Swinnerton-Dyer, H. P. F. (1974). The number of lattice points on a convex curve. J. Number Theory, 6, 128-35. 2,3 Titchmarsh, E. C. (1951). The theory of the Riemann zeta function. Oxford University Press.

5,21

Trifonov, O. (1988). On the number of the lattice points in some two-dimensional domains. Doklady Bulgar. Akad. Nauk, 41, 11, 25-7. 5,18.3 Uchiyama, S. (1972). The maximal large sieve. Hokkaido Maths. J. 1, 117-26. 8.4 Vaughan, R. C. (1977). Sommes trigonometriges sur les nombres premiers. Comptes RenduesAcad. Sci. Paris, A 285, 981-3. 22 Vaughan, R. C. (1981) The Hardy-Littlewood method. Cambridge University Press. 8,12, 23.2 Vinogradov, I. M. (no date; circa 1954). The method of trigonometric sums in the theory of numbers (translated K. F. Roth and A. Davenport). Interscience, London. 5,22 Voronoi, G. (1903). Sur un probli me du calcul des fonctions asymptotiques. J. reine angew. Math., 126, 241-82. 2 Voronoi, G. (1904). Sur une fonction transcendente et ses applications a la sommation de quelques series. Ann. Ecole Norm. Sup. (3) 21, 207-67, 459-533. 6,10, 21.5

490

References

Watt, N. (1988). An elementary treatment of a general Diophantine problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15, 603-14 Watt, N. (1989a). A problem on semicubical powers. Acta Arith., 52, 119-40.

11 11

Watt, N. (1989b). Exponential sums and the Riemann zeta function II. J. London 11,12,17 Math. Soc., 39, 385-404. Watt, N. (1990). A problem on square roots of integers. Periodica Math. Hung., 21, 13 55-64. Watt, N. (1992). A hybrid bound for Dirichlet L-functions on the critical line, in Proc. Amalfi Conf. Analytic Number Theory. University Press, Salerno, 387-92.

16.2,17.3

Weil, A. (1967). Uber die Bestimmung Dirichletscher Reihen durch Funktional10.1 gleichungen. Math. Annalen, 168,149-56. Weil, A. (1971). Dirichlet series and modular forms. Lecture notes in mathematics, 189, 10.1 Springer, Berlin.

Wilton, J. R. (1929). A note on Ramanujan's arithmetic function T(n). Proc. Cam. 10.2 Phil. Soc., 25, 121-9. Wilton, J. R. (1932). Voronoi's summation formula. Quart. J. Math. Oxford (3), 26-32.

10.2

Zaharescu, A. (1987). On a conjecture of Graham. J. Number Theory, 31, 80-7.

1.1

Index

analytic coincidence conditions 315-20, 327, 339-46 approximate functional equation 111-13, 432,

436,441-3,447-8 arc of coincidence 313, 321, 327-8, 341-2 area in intrinsic coordinates 393 Atkin-Lehner newforms 210, 435 Atkinson's summation formula 451

degenerate cases 81 Deligne, P. 210, 425 Deshouillers, J. M. `452 diagonal solutions 236 terms 363 differencing 45-6, 65 step 115-19, 135, 185, 257, 259-60, 363-5, 383, 406, 416, 421-2, 461, 473, 474 Dirichlet approximation theorem 16, 83, 229, 259, 473

Baker, R. C. 473 Balasubramanian, R. 447 Bombieri, E. 2, 3, 4, 125, 158, 160, 289, 467-9, 478, 483

Bombieri-Iwaniec method 142-196, 235-361, 372-383, 479 Borel, E. 7 Brun-Titchmarsh theorem 122-5 Burgess D. A. 135

boundary condition 405 characters 100-101, 135, 236, 362, 432 divisor problem 2, 28, 239-40, 382-83 pigeonhole principle 16, 42-3, 48, 56, 335 series 2, 237, 432-3, 435, 438-441, 447 discrepancy of lattice points 33-41,128-141, 372, 384-8, 398, 474, 479, 483 of sequences 97-8

divide-and-conquer 2, 42-3, 48, 56, 81, 83, 115, 143, 158, 236, 240, 255, 263-4, 307, 311, 447,

Cassels, J. W. S. 116, 242 Cauchy, A. 8 inequality discussed 114-16, 121-2, 125 centre of Farey arc 145, 169, 186, 339 Chaix, H. 41 circle method 143, 152, 255-71, 470-2 see also Hardy, G. H. problem 2, 31, 203 coincidence conditions 156-7, 216, 231, 235, 272, 283-4, 287-346, 349, 416, 426-9, 478,

479

divided differences 46, 65 divisor concentration 240-1, 367 function 29, 203, 237-41, 367 problem 2, 28, 239-40, 382-83 duality of points and lines 73-4, 137-8 of bilinear forms 122, 224

483

detection 119-20, 255 commensurable subgroups 198

Erdos-Turan theorem 96-7 Euler constant 239-40

complementary function 72, 110, 321, 341, 345, 383, 415

product 113, 435, 438-41 Everest, G. 472 exponent pairs 151-2, 369-70 exponential integrals defined 87 evaluated 104-14 exponential sums bounded 99,353-61,365-71,415-22 defined 87 non-standard 475

congruence family 336-9,349,357,361-3,392, 478

continued fraction 16-19, 398 Corput, J. G. van der 4, 41, 43, 93, 116, 125 iteration 43, 62, 116-17, 135, 152, 383, 406, 474

lemmas 87-120, 221-2, 225, 448 counting squares 1-2, 25-31, 41, 131, 167 cusp forms 200 Davenport, H. 4, 101, 453

Faltings, G. 235

Index

492

families of solutions 241-48, 249-51, 272-83 of sums 179, 194, 219-20, 300-306, 333-4, 353, 357-61, 393-7, 410-13, 420, 424-9, 458, 478

Farey arcs 2, 143, 167-70, 185-7, 208-9, 255, 336-7, 372, 425-30, 445, 470, 473, 479-80, 483 sequence 8-10, 19-23, 208, 346, 439, 470, 477

see also major arcs, minor arcs

Fejer kernel 93-4 Filaseta, M. 45, 470 First Derivative Test stated 88, 104, 113-14, 448

First Spacing Problem 3, 56, 158, 235-85, 307,

372,406,471,476-7 flanks of major arcs 151 Ford circles 10 Fortean function 312, 324, 327, 342, 343 Fouvey, E. 475, 477

gardening 152,166,183-4,195-6,225,228-30, 412

Gauss circle problem 2, 31, 203 sums 100-4, 149, 162 Gel'fond, A. 0. 41, 135, 452 generating function 198, 438, 470 Gershgorin's eigenvalue bound 224 Good, A. 436, 439 Graham, R. L. 8 Graham, S. W. 4, 432 with Kolesnik, G. 2, 43, 116, 125, 152, 297, 383, 474

Halasz, G. 433 Halberstam, H. 122 Hall, R. R. 240, 399 Hardy, G. H. 2, 31, 122, 143, 203, 256, 439, 453

Hardy-Littlewood circle method, see circle method Harman, G. 473 Heath-Brown, D. R. 2, 4, 125, 185, 406, 444, 451, 464

height 7 Herz, C. S. 128, 474 Hlawka, E. 128, 405, 474 Hobson, E. W. 130 Hooley, C. 4 divisor concentration 241-2, 367

intrinsic equation of curve 24-5, 384, 388-393 inversion step 108-10, 116-17, 149, 176, 191-2, 193, 383, 422, 474-5 Ireland, K. 210

iteration between rational points and lines 76-80 by major arc method 166, 413-15 of points close to curves 43, 48-53 of resonance curves 370, 415-22 van der Corput's 43, 62, 116-17, 137, 154, 383, 406, 474 Ivic, A. 4, 440 Iwaniec, H. 2, 3, 4, 125, 167, 372, 451, 471, 475, 477; see also

Bombieri-Iwaniec

Jarnik, W. 30-33, 57 Jeffreys, H. and B. S. 150, 177, 203, 441, 482 Jordan curve theorem 25 Jutila, M. 2, 4, 208, 225, 284, 423, 435, 436, 437, 451

Karatsuba, A. A. 125, 160, 451 Kendall, D. G. 135, 474 Kloosterman refinement 471, 479

sums 481-3 Koebe's function 440 Kolesnik, G. 2, 4, 255, 297, 339, 416, 474, 478, see also Graham S. W. Kratzel, E. 4, 43, 474, 475 Kuznietsov, N. V. 290, 439, 481-83 L-function 435-7, 440-1, 451 Landau, E. 31 large sieve 3, 120-125, 126, 166, 214-20, 224-5,372,470 applied 152-8, 178-84, 193-6, 231, 329, 475-6

lemmas 120-25, 224-5 large values results 126, 431-2 Lax, P. D. 440 limitation results 126, 248-9, 334-5, 474 Lindelof hypothesis 441 Linnik, Yu. V. 41, 122, 124, 135, 454 Liu, H.-Q. 452, 475 Logan, I. 398 magic matrix 289-308, 320-1, 329-335, 415-18, 423-8, 430 major arcs in Bombieri-Iwaniec method 148-51, 160-66, 168-72, 186-8, 377, 409, 480 flanks of 151 in Hardy-Littlewood method 143, 255, 256-7, 262-68, 270 long 160-66, 356, 413-15, 450 for S-suits 473 Sargos splines 469 as sides of polygon 55, 67-72 Mangoldt, H. von 438, 453, 464 Matthews, K. R. 125 mean-to-max 117-18, 122, 135, 136, 185, 410

493

Index

Mellin transform 91, 117-18, 181-2, 195, 438-40 Minkowski, H. 242 minor arcs in Bombieri-Iwaniec method 143-9, 151, 168-70, 172-8, 186-7, 188-93, 336, 377, 410, 414, 480

in Hardy-Littlewood method 143, 255, 256-62, 270 Sargos splines 469

as sides of polygon 55, 67 MSbius function 13-14, 63-4, 123-5, 439, 454 transformation 199 Modification One 147, 149, 168, 172, 315 Two 147, 149, 156, 169, 181, 186, 195, 351, 352, 374, 376 Three 169

reflection step, see inversion step resonance curves 320-21, 345, 415-19 Riemann hypothesis 15, 210, 435-6, 439-40, 447

zeta function 2, 3, 64, 111-13, 160, 237, 432, 435, 438-451

Riesz interchange 218, 246, 267, 330-2, 405, 457

Rivat, J. 452, 475 Rosen, M. 210 row-of-glasses function 95 row-of-teeth function 2, 33, 93, 131, 167, 379-81 rounding error function 2, 33, 93, 131, 167, 379-81 sums 93-8, 131, 167, 363, 372, 379-83, 385-8,397-405

modular group 11-12, 197, 289, 439-40, 481, 483 forms 2, 3, 197-9, 256, 290, 435, 451, 481-83

Sargos splines 469

monomial function 50-2, 117, 165, 369, 414, 416,473,475-7 Montgomery, H. L. 115, 213, 455 Mordell, L. J. 440 Moreno, C. J. 210, 425, 437 Motohashi, Y. 4 Mozzochi, C. J. 2, 4, 167, 372

Second Spacing Problem 3, 158, 287-346, 363, 372, 476, 483 Selberg, A. 4, 481 sieve 122, 125, 439

Nowak, W. G. 4, 135, 474

Sierpinski, W. 41, 167, 372 sieve

trace formula 439-40 zeta function 439-40 Shahidi, F. 210, 425, 437 short interval means 118-19, 221-4, 226-7,

408-10,435,443-7 general 454

O'Hara, F. 398 partial summation lemmas 89-93, 181-2, 225, 408

Phillips, R. S. 440 Piatecki-Shapiro theorem 452, 463, 475 Pila, J. 467-9 Pisot, Ch. 116 Poisson summation formulae 93-104 used 2,103,109,110-12,149-50,173-8,

large, see large sieve Selberg's 122, 125, 439 simple asymptotic 63 simple asymptotic sieve 63 Simpson's rule 397 stakanchik 95 stationary phase integrals 105-8, 133-4, 150, 177, 212, 263

Swinnerton-Dyer, H. P. F. 4, 30, 43, 45-6, 49, 50, 54-62, 197, 467

189-93, 257, 260, 263-4, 320, 363, 381,

383,409,419,432,438,474-5 polynomial approximation 2, 142-6, 161, 173,

188-9,384,469 prime numbers 1, 15, 122-5, 438-9, 447, 452-64, 473

Tenenbaum, G. 240, 399 trapezium rule 31, 98, 128-9, 397 Trifonov, O. 4, 470 Type I double sums 115, 455-60 Type II double sums 115, 213, 455, 459-62, 475

Rademacher's binary decomposition 180 Ramachandra, K. 446 Ramanujan's hypothesis: bounds 210 hypothesis: multiplicativity 440 sum 13-14, 123, 125 work on partitions 143, 256 Rankin R. A. 4, 200, 210, 425, 433, 436, 451

reference fractions 19-20, 308, 312, 329-30, 332

Uchiyama, S. 180 uniform Diphantine approximation 19-23, 221 uniform distribution 12-15

van der Corput, see Corput, J. G. van der Vaughan, R. C. 115, 143, 152, 256, 454, 455, 464, 471

Index

494

Vinogradov, I. M. 4, 41, 93, 143, 152, 256, 454,

Wright, E. M. 453

471

Voronoi, G. 4, 41, 128 polygon 2, 40-1, 167, 372 summation formula 128, 203, 451

x-vectors 154, 178, 193, 231 y-vectors 154, 178, 194, 231

Waring's problem 3, 152, 471 Watt, N. 2, 4, 20, 241, 246, 272, 297 Weil, A. 199, 481 Weyl's criterion 13, 96-7 Wilton summation 199, 200-10, 451 translated 440 used 210-13

Zaharescu, A. 8 zeta functions 440-1 Minakshisundaram-Pleijel 440 Riemann 2, 3, 64, 111-13, 160, 237, 432, 435, 438-451 Selberg 439-40