The Distribution of Prime Numbers: Large Sieves and Zero-density Theorems (Oxford Mathematical Monographs)

O X F O R D M A T H E M A T IC A L M O N O G R A P H S

M E R O M O E P H IC FU N C TIO N S By w.

k. haxm an.

1963

T H E T H E O R Y OF L A M I N A R B O U N D A R Y L A Y E R S IN C O M P R E SSIB L E F L U ID S By K. SIEWASTSON. 1964 C LA SSIC A L H A R M O N IC A N A L Y S I S A N D L O C A L L Y COMPACT GROUPS By H. EBITER. 1968 Q U A N T U M - S T A T I S T I C A L F O U N D A T I O N S OF C H E M IC A L K IN E T IC S By s.

golden.

1969

CO M P LE M E N TA R Y V A R IA T IO N A L PR IN C IP LE S By

a . m. a e t h u b s .

1970

V A R IA T IO N A L P R IN C IP L E S IN H E A T T R A N SF E R By MAURICE A. BIOT. 1970 P A R T IA L W A V E A M P L IT U D E S AND R ESO N AN CE POLES By J. H a m i l t o n and

b. tromborg.

1972

THE DISTRIBUTION OF PRIME NUMBERS Large sieves and zero-density theorems BY

M. N . H U X L E Y

OXFORD AT TH E CLAREN D O N PRESS 1972

Oxford University Press, Ely House, London W. 1 G LA SGOW C APE TO W N D ELHI

NEW YORK

IB A D A N

BOM BAY

TORONTO

N A IR O B I CALCU TTA

K U A L A LU M PU R

M E LB O U R N E

D A K ES SALAAM MADRAS

SIN G A PO R E

W E LLIN G TO N

LUSAKA

KARACHI HONG K O N G

A D D IS A B A llA

LA H O R E TO K Y O

© Oxford University Press 1072

Printed in Great Britain at the University Press, Oxford by Vivian Midler Printer to the University

DACCA

TO T H E M E M O R Y OF P R O F E S S O R H. D A V E N P O R T

PREFACE T h i s book has grown out o f lectures given at Oxford in 1970 and at

University College, Cardiff, intended in each ease for graduate students as an introduction to analytic number theory. The lectures were based on D avenport’ s Multiplicative Number Theory, but incorporated simpli­ fications in several proofs, recent work, and other extra material. Analytic number theory, whilst containing a diversity o f results, has one unifying method: that o f uniform distribution, mediated b y certain sums, which m ay be exponential sums, character sums, or Dirichlet polynomials, according to the type o f uniform distribution required. The study o f prime numbers leads to all three. Hopes o f elegant asym­ ptotic formulae are dashed b y the existence o f complex zeros o f the Riemann zeta function and o f the Dirichlet L-functions. The primenumber theorem depends on the qualitative result that all zeros have real parts less than one. A zero-density theorem is a quantitative result asserting that not many zeros have real parts close to one. In recent years many problems concerning prime numbers have been reduced to that o f obtaining a sufficiently strong zero-density theorem. The first part o f this book is introductory in nature; it presents the notions o f uniform distribution and o f large sieve inequalities. In the second part the theory o f the zeta function and L-functions is developed and the prime-number theorem proved. The third part deals with large sieve results and mean-value theorems for L-functions, and these are used in the fourth part to prove the main results. These are the theorem o f Bombieri and A. I. Vinogradov on primes in arithmetic progressions, a result on gaps between prime numbers, and I. M. Vinogradov’s theorem that every large odd number is a sum o f three primes. The treatment is self-contained as far as possible; a few results are quoted from Hardy and W right (1960) and from Titchmarsh (1951). Parts o f prime-number theory not touched here, such as the problem o f the least prime in an arithmetical progression, are treated in Prachar’s Primzahlverteilung (Springer 1957). Further work on zero-density theorems is to be found in Montgomery (1971), who also gives a wide list o f references covering the field. M. N. H. Cardiff 1971

CONTENTS

PART

I.

INTRODUCTORY

RESULTS

1. Arithmetical functions

1

2. Some sum functions

6

3. Characters

10

4. P olya’s theorem

14

5. Dirichlet series

18

6. Schinzel’s hypothesis

23

7. The large sieve

28

8. The upper-bound sieve

32

9. Franel’s theorem

36

P A R T II.

THE

PRIM E-NUM BER

THEOREM

10. A modular relation

40

11. The functional equations

45

] 2. HadamarcPs product formula

50

13. Zeros o f £(s)

55

14. Zeros o f £{s, x)

58

15. The exceptional zero

61

16. The prime-number theorem

66

17. The prime-number theorem for an arithmetic progression

70

PART

III.

THE

NECESSARY

TOOLS

18. A survey o f sieves

73

19. The hybrid sieve

79

20. An approximate functional equation (I)

84

21. An approximate functional equation (II)

89

22. Fourth powers o f ^-functions

93

X

CONTENTS

PAET IV.

ZEROS AND

PRIM E N U M B E R S

23. Ingham ’s theorem

98

24. Bom bieri’s theorem

103

25. I. M. Vinogradov’s estimate

107

26. I. M. Vinogradov’s three-primes theorem

110

27. Halasz’s method

114

28. Gaps between prime numbers

118

N O T A TIO N

123

B IB LIO G R A P H Y

124

IN D E X

127

PART

I

Introductory Results

1

A R ITH M E TIC A L

FU N CTION S

An Expotition . . . means a long line of everybody

I. 110 T h i s chapter serves as a brief resume o f the elementary theory o f prime

numbers. A positive integer m can be written uniquely as a product o f primes

m _

( 1,1)

where t h e ^ are primes in increasing order o f size, and the ai are positive integers. W e shall reserve the letter p for prime numbers, and write a sum over prime numbers as 2 ancl a product as JT- The p roof o f p i> unique factorization rests on E uclid’s algorithm that the highest com ­ mon factor (m, n) o f two integers (not both zero) can be written as (m,n) = m u + n v,

( 1.2)

where u, v are integers. W e use (m, n) for the highest common factor and \m, n\ for the lowest common multiple o f two integers where these are defined. Let # be a positive integer. Then the statement that m is congruent to n (m od#), written m = n (m od#), means that m —n is a multiple o f q. Congruence m od q is an equivalence relation, dividing the integers into q classes, called residue classes m od#. A convenient set o f representa­ tives o f the residue classes mod q is 0, 1, 2,..., q —I. The residue classes m od# form a cyclic group under addition, and the exponential maps m -> eQ(am), where a is a fixed integer, and

(1.3)

2

IN T R O D U C T O R Y RESULTS

e(a) = exp(27ria),

eg(a) = exp(27ria/g),

1.1

(1.4)

are homomorpliisms from this group to the group o f complex numbers o f unit modulus under multiplication. There are q distinct maps, corre­ sponding to a = 0, 1, 2,..., q—■1. They too can be given a group structure, forming a cyclic group o f order q. They have the important property

( i -5> where the summation is over a complete set o f representatives o f the residue classes m od# (referred to briefly as a complete set o f residues m od#). I f on the left-hand side o f eqn (1.5) we replace to b y t o + 1 , the sum is still over a complete set o f residues, but it has been multiplied b y ea(a), which is not unity unless a = 0 (m od#). The sum is therefore zero unless a = 0 (m odg), when every term is unity. Interchange o f a and to leads to a corresponding identity for the sum o f the images o f m under a complete set o f maps (a — 0, 1,..., q— 1). These identities arise because the images lie in a multiplicative not an additive group. From E uclid’ s algorithm comes the Chinese remainder theorem: if to, n are positive integers and (m, n) = 1, then any pair o f residue classes a (mod to) and b (mod?i) (which are themselves unions o f residue classes modwm) intersect in exactly one class c (modtow), given by c = bmu-\-anv (mod tow)

( 1,6)

in the notation o f eqn (1.2). N ow let /(to ) be the number o f solutions (ordered sets (x1,...,x r) o f residue classes) o f a set o f congruences gi{xi,...,xr) = 0 (m odto),

(1.7)

where the gi are polynomials in xr with integer coefficients. When (m ,n) = 1, gi(x1>...,x r) is a multiple o f mn if and only if it is a multiple both o f to and o f n. Hence f(m n ) = f(m )f(r i)

whenever (m ,n ) = 1.

(1.8)

Equation (1.8) is the defining property o f a multiplicative arithmetical function. An arithmetical function is an enumerated subset o f the complex numbers, that is, a s e q u e n c e /( l ) ,/( 2),... o f complex numbers. The property

f(m n )= f(m )f(n )

(1.9)

for all positive integers m and n seems more natural; if eqn (1.9) holds as well as (1.8) th e n /(to ) is said to be totally multiplicative, but (1.8) is the property fundamental in the theory.

1.1

A R IT H M E T IC A L FU N C T IO N S

3

The Chinese remainder theorem enables us to construct more compli­ cated multiplicative functions. W e call a residue class a (mod#) reduced if the highest common factor (a, q) is unity. A sum over reduced residue classes is distinguished by an asterisk. W ith this notation we introduce Euler’s function