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

OXFORD MATHEMATICAL MONOGRAPHS

MEROMORPHIC FUNCTIONS By W. K. HAYMAN. 1963

THE THEORY OF LAMINAR BOUNDARY LAYERS IN COMPRESSIBLE FLUIDS By K. STEM WARTSON. 1964

CLASSICAL HARMONIC ANALYSIS AND LOCALLY COMPACT GROUPS By H. REITER. 1968

QUANTUM. STATISTICAL FOUNDATIONS OF CHEMICAL KINETICS By S. GOLDEN. 1969

COMPLEMENTARY VARIATIONAL PRINCIPLES By A. M. ARTHURS. 1970

VARIATIONAL PRINCIPLES IN HEAT TRANSFER By MAURICE A. BIOT. 1970

PARTIAL WAVE AMPLITUDES AND RESONANCE POLES By J. HAMILTON and B. TROMBORG. 1972

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

M. N. HUXLEY

OXFORD AT THE CLARENDON PRESS 1972

Oxford University Press, Ely House, London W. 1 GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON CAPE TOWN IBADAN NAIROBI DAR ES SALAAM LUSAKA ADDIS ABA11A DELHI BOMBAY CALCUTTA MADRAS KARACHI LAHORE DACCA KUALA LUMPUR SINGAPORE HONG KONG TOKYO

© Oxford University Press 1972

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

TO

THE MEMORY OF

PROFESSOR H. DAVENPORT

PREFACE THIS book has grown out of lectures given at Oxford in 1970 and at University College, Cardiff, intended in each case for graduate students as an introduction to analytic number theory. The lectures were based on Davenport's 11Multiplicative Number Theory, but incorporated simpli-

fications in several proofs, recent work, and other extra material. Analytic number theory, whilst containing a diversity of results, has one unifying method: that of uniform distribution, mediated by certain sums, which may be exponential sums, character sums, or Dirichlet polynomials, according to the type of uniform distribution required. The study of prime numbers leads to all three. Hopes of elegant asymptotic formulae are dashed by the existence of complex zeros of the Riemann zeta function and of 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 of obtaining a sufficiently strong zero-density theorem. The first part of this book is introductory in nature; it presents the notions of uniform distribution and of large sieve inequalities. In the second part the theory of 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 of 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 of three primes. The treatment is self-contained as far as possible; a few results are quoted from Hardy and Wright (1960) and from Titchinarsh (1951). Parts of prime-number theory not touched here, such as the problem of 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 of 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 4. Polya's theorem

10

5. Dirichlet series

18

6. Schinzel's hypothesis

23

7. The large sieve 8. The upper-bound sieve 9. Franel's theorem

28

14

32

36

PART II. THE PRIME-NUMBER THEOREM

10. A modular relation 11. The functional equations 12. Hadamard's product formula 13. Zeros of f(s)

40

1.4. Zeros of e(s, X)

58

15. The exceptional zero 16. The prime-number theorem 17. The prime-number theorem for an arithmetic progression

61

45 50 55

66 70

PART III. THE NECESSARY TOOLS

18. A survey of sieves

73

19. The hybrid sieve 20. An approximate functional equation (I) 21. An approximate functional equation (II) 22. Fourth powers of L-functions

79

84 89 93

CONTENTS

s

PART IV. ZEROS AND PRIME NUMBERS

23. Ingham's theorem 24. Bombieri's theorem 25. I. M. Vinogradov's estimate 26. I. M. Vinogradov's three-primes theorem 27. Halasz's method 28. Gaps between prime numbers NOTATION

BIBLIOGRAPHY

INDEX

PART I

Introductory Results

1

ARITHMETICAL FUNCTIONS An Expotition ... means a long line of everybody

I. 110

THIS chapter serves as a brief resume of the elementary theory of prime

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

(

m = pi'p2E ..* prr,

1.1

)

where the pi are primes in increasing order of size, and the a1 are positive

integers. We shall reserve the letter p for prime numbers, and write a sum over prime numbers as and a product as H. The proof of v

P

unique factorization rests on Euclid's algorithm that the highest common factor (m, n) of two integers (not both zero) can be written as

(m, n) = mu+nv,

(1.2)

where it, v are integers. We use (m, n) for the highest common factor and [rn, n] for the lowest common multiple of two integers where these are defined. Let q be a positive integer. Then the statement that m is congruent ton (mod q), written m - n (mod q), means that m-n is a multiple of q. Congruence mod q is an equivalence relation, dividing the integers into q classes, called residue classes mod q. A convenient set of representatives of the residue classes mod q is 0, 1, 2,..., q- 1. The residue classes mod q form a cyclic group under addition, and the exponential snaps m - eq(am), where a is a fixed integer, and

(1.3)

2

INTRODUCTORY RESULTS

e(a) = exp(2iria),

eq(a) = exp(27ria/q),

1.1

(1.4)

are hoinomorphisms from this group to the group of complex numbers of unit modulus under multiplication. There are q distinct maps, corresponding to a = 0, 1, 2,..., q-1. They too can be given a group structure, forming a cyclic group of order q. They have the important property

I e ama q(

mmodq

)

rq 0

if a- 0 (mod q), if not,

(

15

)

where the summation is over a complete set of representatives of the residue classes mod q (referred to briefly as a complete set of residues mod q). If on the left-hand side of eqn (1.5) we replace m by m+1, the sum is still over a complete set of residues, but it has been multiplied by eq(a), which is not unity unless a - 0 (mod q). The sum is therefore zero unless a - 0 (mod q), when every term is unity. Interchange of a and m leads to a corresponding identity for the sum of the images of m under a complete set of maps (a = 0, 1,..., q- 1). These identities arise because the images lie in a multiplicative not an additive group. From Euclid's algorithm comes the Chinese remainder theorem: if m, n are positive integers and (m, n) = 1, then any pair of residue classes a (mod m) and b (mod n) (which are themselves unions of residue classes modmn) intersect in exactly one class c (modmn), given by

c - bmu+anv (modmn)

(1.6)

in the notation of eqn (1.2). Now let f(m) be the number of solutions (ordered sets (x1,..., x,,) of residue classes) of a set of congruences (1.7) gi(xl...... ,,) - 0 (mod m), where the gi are polynomials in x1,..., x,, with integer coefficients. When (m, n) = 1, gi(xi,..., x,.) is a multiple of mn if and only if it is a multiple both of m and of n. Hence

f (inn) = f (m) f (n) whenever (m,%) = 1.

(1.8)

Equation (1.8) is the defining property of a multiplicative arithmetical

function. An arithmetical function is an enumerated subset of the complex numbers, that is, a sequence f (1), f (2),... of complex numbers.

The property

f (mn) = 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) then f (m) is said to be totally multiplicative, but (1.8) is the property fundamental in the theory.

1.1

ARITHMETICAL FUNCTIONS

3

The Chinese remainder theorem enables us to construct more complicated multiplicative functions. We call a residue class a (mod q) reduced if the highest common factor (a, q) is unity. A sum over reduced residue classes is distinguished by an asterisk. With this notation we introduce Euler's function p(m) by

cp(ma) _ " 1.

(1.10)

amodm

To show that y(m) is multiplicative, we must verify that in eqn (1.6) (c, man) = 1 if and only if both (a, m) and (b, n) are unity. Equation (1.6) implies also that Ramanujan's sum cq(m) _ :ET eq(ama)

(1.11)

auiodq

is multiplicative in q for each m. We see this if we write a = a2 g12G2+a1 q2 u1,

(1.12)

(1.13) g12G2+g2u1 = 1; where note that u1 (mod q1) and u2 (mod q2) are reduced residue classes, that

Y* eq,(a2u2)eq,(aiui), c'hg2(rn) = 1'' asmodgs aluiodq,

(1.14)

and that al u1 runs through a complete set of reduced residues mod ql when a1 does so. Two examples follow of totally multiplicative arithmetical functions.

The first is f (m) = mas, where s is a complex variable

s = a+it,

(1..15)

(1.16)

a and t being real. This notation is traditional among number theorists. To introduce our second example we note that the reduced residues

mod q (algebraically the invertible elements in the ring of integers mod q) form under multiplication an Abelian group of order p(q). By considering the images of the generators of this group, we can see that from this group to the group of complex numbers of unit modulus under multiplication there are p(q) maps x with the homomorphism property x(mmn) = x(m)x(n).

(1.17)

These include the trivial map for which x(m) = 1 for each reduced class m.

We turn these maps into arithmetical functions by defining (1.18) x(m) = 0 if (m, q) > 1. With this definition, eqn (1.17) still holds. We have now assigned a complex number to each residue class mod q. Hence we have constructed

INTRODUCTORY RES1YLTS

4

1.1

a totally multiplicative periodic function, which is called a Dirichlet's character mod q, or more briefly a character. Characters can be defined as those totally multiplicative functions that are periodic. Since negative integers also belong to well-defined residue classes mod q, we can speak of x(m) when m is a negative integer; in particular, we shall refer to

x(-1).

It is possible to build new multiplicative functions from old. We say that d divides m, written d I rn, when the integer rn is a multiple of the positive integer d; another paraphrase is 'd is a divisor of m'. (Note that the divisors of - 6 are 1, 2, 3, 6.) Now let f (rn) and g(m) be multiplicative.

Then so are the arithmetical functions h(m) = f (m)g(m),

h(m) = If (d), dbn

h(m) = If (d)g(m/d).

and

(1.21)

d1 m

We shall consider eqn (1.21), since (1.20) is a special case, and (1.19) is evident. When (m, n) = 1, the divisor d of mn can be written uniquely

as d = ab, where a I m and b n, and (a, b) = 1. Hence h(mn) =

f (d)g(mn/d)

dIn _ I If (ab)g(mn/ab) alm bin

_ Y_ If (a) f (b)g(m/a)g(n/b),

(1.22)

alm bin

which is h(m)h(n) as required. Thus

d(m) = 11,

(1.23)

din

the number of divisors of m, and

a(m) =I d,

(1.24)

dim

the sum of the divisors of m, are multiplicative functions.

We can invert eqn (1.20) and return from h(m) to f(d) by using 1116bius's multiplicative function µ(m), defined by

µ(1) = 1

µ(p) = -1 for prunes p µ(pa) = 0

for prime powers pa with a > 1

(1.25)

ARITHMETICAL FUNCTIONS

1.1

5

If the positive integer 9n factorizes according to (1.1), then µ(d) _ +p(pi)+p(p%)+...-}-µ(pa')} d1m

(1-1)=0,

1.26)

unless in = 1, when the product in eqns (1.1) and (1.26) is empty. We have now proved the following lemma. LEMMA. If in is a positive integer, then

if In = 1, if m > 1. From the lemma we have the corollary: d1m

d=

1

0

(

1.27 )

COROLLARY. If h(m) and f(m) are related by eqn (1.20), then

f (n) = I µ(in)h(n/m),

(1.28)

mm

and if eqn (1.28) holds then so does eqn (1.20).

To prove the corollary we substitute as follows. j µ.(mn)h.(n/m) _ ,j!,(mm) I f (d) mIn

mjn

dl(n/m)

=f(d) mi(I

t

k(m)

(1.29)

when we interchange orders of summation. The inner sum is zero by eqn (1.27), unless d = n, when only one term f (d) remains. The converse is proved similarly. We can also define an additive function to be an arithmetical function f (m) with (1.30) f(mn) = f (9n) +f (n) when (in, n) = 1.

The simplest examples are log m and the number of prime factors of in. There are useful arithmetical functions that are neither multiplicative nor additive. We shall make much use of 11(m), given by !1(m) = (logp if in is a prime power pa, a > 1, (1.31) 0 if in is not a prime power.

It satisfies the equation

I!1(d) = login.

(1.32)

dim

We could have used eqn (1.32) to define !1(m) and recovered the definition (1.31) by Mdbius's inversion formula (1.28).

2

SOME SUM FUNCTIONS THE study of the sum functions of arithmetical functions is important

in analytic number theory. For instance, we shall treat many of the properties of prime numbers by using the sum function

O(x) _ I A(rn).

(2.1)

nt6 x

Our object is to express the sum function as a smooth main term (a power of x or of log x, for example) plus an error term. In place of the cumbersome

(2.2)

If (x) I = O(g(x)),

we shall often write

f (x) < g(x),

(2.3)

and other asymptotic inequalities similarly. Some sum functions can be estimated by writing the arithmetical function as a sum over divisors and rearranging. In this chapter we shall give examples of this method. From the theory of the logarithmic function we borrow the relation 114

n-log( " nt=1 `1

11

1))

`

wh ere y is a constant lying between formula

Y+0(T,

=

),

(2.4)

and 1. We deduce the useful

1

log(117+1)+y+O(-31-1). ttt=1

(2.5)

r!L

Our first example is an asymptotic formula for (2.6)

(P(x) _ I On) In<X

as x tends to infinity. Since p(m) is the number of integers r with 1 < r < m and (r,rn) = 1, eqn (1.27) gives

_ (1 d1m

a",

0

if if not.

1, (

2.7

)

SOME SUM FUNCTIONS

1.2

Hence

and so

7

9n

p(rn) _ r=1 d m µ(d) (P(x)

_ m<x r-1 cc ?n FL(d) 'In

d

d5x

d5x

1

m<x

r=1

m=0(modd) r=O(modd)

I

/x(d) m

o(mo

and dd)

n<x/d

d<x

where we have written n for in/d, and the sum over n is 'E(x/d)2+ 0(x/d).

We now write eqn (2.9) as 1

(P(x)

µ(d)

2

<x -

(2.10)

2d

`+0(a,))

d2 Fi()+ 0C

x.22

d

d

11

dam,x

dJ

OD

d)+ 0(x)

0(xlogx)

(2.11) = cx2+0(xlogx), where the constant c can be evaluated; in the notation of Chapter 5 it

which can be shown to be 37r -2. The size of the error term is highly satisfactory: when x is a prime, O(x) has a discontinuity x-1, and the actual error in (2.11) is > x infinitely often. Even so, the upper bound 0(x log x) for the error term in (2.11) has been improved a little. When we first apply the same method to (2.12) D(x) = Y_ d(m), is

m<x

we find that

D(x) = I

1

m5x rim

\r+ 0(1)l = x(logx+y+O(x-1))+O(x) r<x

(2.13) = xlogx- -O(x), by (2.5). The error term here is much larger in proportion to the main 8535185

B

INTRODUCTORY RESULTS

8

1.2

term than in (2.12); in fact a little cunning enables us to improve (2.13). We let y be the positive integer for which y2 < x < (y+1)2, and write m = qr in eqn (2.12), so that D(x)

gr5x

=1 1 1+1 1 1_y- 11 45v r-<xl4 rbvgSx/r

4'S?/ , y

= 2 QSU 1 (xlq+o(1))-y2 = xlogx+(2y-1)x+0(x1),

(2.14)

when we substitute (2.5) and the value of y. Dirichlet's divisor problem is to improve the exponent of x in the error term of (2.14). I. M. Vinogradov (1955) has shown that the error term in (2.14) is < x1log2x by elementary arguments. Van der Corput and others have obtained estimates < xS with values of 8 a little less than s ; their method involves writing the error in (2.14) as a contour integral.

On the other side, it has been shown that the error term is > xa for each 8 < 1. The limitation on the accuracy of (2.14) is not so easily explained as that of (2.11): it is not difficult to show that d(m) <ml (2.15) for each t > 0, and so the error in (2.14) must often be very much greater than any individual step in the value of D(x). To illustrate more difficult examples, we consider (2.16)

I d2(m)b)Im.

rn<x

Let dr(m) be the number of ways of writing the positive integer m as a product of r positive integers (so that d(m) = d2(r)Z)). By induction on (2.14),

G dr-1(v) Ui)

X

'v loge-lx (, _ 1) (

-f -

O(x log''-2x),

(2.17)

and by partial summation d4(m) ?n<x

in

D4(x) m5x

ln(rn+1)

+

1 1093X

l x l

= 241og4x+0(log3x).

(2.18)

SOME SUM FUNCTIONS

1.2

9

We have chosen to compare d2(m) with d4(m) since, when m is a prime power pa, d4(7n) = j(a+1)(a+2)(a+3), (2.19)

which is equal to d2(rn) if a = 0 or 1. The next step is to find a function b(m) for which (2.20) d2(m) = d4(u)b(v). UV=24

Since

*(a+1)(a+2)(a+3)-j(a-1)a(a+1) = (a+1)2,

(2.21)

the choice b(1) = 1, b(p2) = -1, b(pa) = 0, for prime powers pa with a not zero or two, satisfies eqn (2.20) when in is a prime power. If we complete the definition of b(m) by making it multiplicative, then (2.20) holds for all m. The choice is thus b m = J14(n) if m = n2, 2.22 if m is not a perfect square. We now complete the proof. Equations (2.20) and (2.22) give (

)

0

d2(m)

,n

-

(

)

d(u)b(v) 1-4

UV

uv5x

(2.23) ta5x

t2

uSx/ta

When we substitute (2.18) and the value 67r-2 of have

mx d2mm)

00

1.t(t)t-2 into this, we

1

- (+o1)log4x.

(2.24)

We require (2.24) and upper estimates for similar sums in the later work.

Any < estimate for a sum involving divisor functions that we quote will be a corollary-of (2.14) or of (2.24), possibly using partial summation. The method we employ in this chapter can be summarized as follows.

To work out a general sum function

F(x) = I f (m),

(2.25)

m<- x

we try to write

f (in) = E a(u)b(v), uv=m

(2.26)

where we have an asymptotic formula for the sum function of the a(m), and b(m) is in some sense smaller than a(m). A necessary condition is that b(p) = o (ja(p) 1).

(2.27)

Our determination of O(x) depended on the equation

On) _ uv=m I u/)-(V)' which is of the form (2.26); and eqn (2.20) is clearly of this form.

(2.28)

3

CHARACTERS THE reduced residue classes mod q form a group under multiplication, and the characters modq correspond to the maps m - X(m) from this group to the group of complex numbers of unit modulus under multiplication. We define a group operation on the set of the p(q) characters mod q: the product Xi X2 of two characters Xi and X2 mod q is the map m - X1(m)X2(m). The unit of this group is the trivial character modq.

When the group of reduced residues and the group of characters are each expressed as a direct sum of cyclic groups, we can see that they are isomorphic and that the homomorphisms of the group of characters to the complex numbers are given by X - X(m), where m runs through the reduced residues modq. Two finite Abelian groups related in this way are said to be dual. We have already seen that e,2(am)

(q

t0

rmmodq

if a if a

0 (mod q),

0 (modq),

(3.1)

for the maps m --> eq(am) from the group of residue classes m (mod q)

under addition to the complex numbers of unit modulus. We have similar results for the characters X(m) modq:

I On) _ (cp(q) (0

namodq

if X is trivial, if X is non-trivial.

(3.2)

Here X(m) is non-zero when the residue class m (mod q) is reduced, and thus when m is a member of the group of reduced residue classes under

multiplication. Now eqn (3.1) has a dual interpretation, in terms of residue classes a (mod q) and maps a -+ eq(am). This corresponds to I X(m) = fcp(q) if m 1 (modq), (3.3) 0 if m 1 (modq). Xmddq The proof of eqn (3.2) runs as follows. If X is trivial, then X(m) = 1 when (m, q) = 1 and 0 otherwise, so that the sum on the left-hand side

of (3.2) is the definition (1.10) of p(q). If X is non-trivial, there is an integer r for which (r, q) = 1 but X(r) is not unity. We multiply the sum on the left of eqn (3.2) by X(r); this gives

X(mr), where m runs through

1.3

CHARACTERS

11

a complete set of residues mod q. Since (r, q) = 1, mnr also runs through

a complete set of residues modq, and the sum is unchanged. This 1 if the sum is not zero. Similarly, contradicts the choice of r with X(r) in eqn (3.3) if in is not congruent to unity mod q there is a character X, 1. When we multiply the left-hand side of (3.3) by Xi(m), with X1(m) X(m)X1(m) is a sum over all characters mod q, and again this sum

must be zero, for we have multiplied by a constant that is not unity but have succeeded only in permuting the terms of the sum. Of course, eqns (3.1), (3.2), and (3.3) are essentially special cases of a theorem on dual finite Abelian groups. An important notion is the propriety of characters. Let q2 be a multiple of qr, and Xr a character mod q1. The group of reduced residue classes mod q2 maps homomorphically onto the corresponding group mod q1, and we define a character X2 mod q2 by the equation f Xi(m)

X2(m) = 10

if (in, q2) = 1, if (m, q2) > 1.

(3.4)

We note that Xr and X2 are different arithmetical functions. If q1 = 3, q2 = 6, and Xr takes the values

1, -1, 0, 1, -1, 0

(3.5)

for in = 1, 2, 3, 4, 5, 6, then X2 takes the values

1, 0, 0, 0, -1, 0,

(3.6)

since X2 is zero when rn is a multiple of 2 as well as when in = 0 (mod 3).

When X2 is constructed by eqn (3.4), we say that Xi modgr induces X2 mod q2, and, if q2

q1, that X2 mod q2 is imnproper. A proper character

mod q (also called a primitive character) is one that is not induced by a character mod d for any divisor d of q other than q itself. The smallest f for which a character Xr mod f induces X mod q is called the conductor of X. The customary letter f is the initial of a German word for a tram conductor. We shall now discuss Gauss's sum T(X), defined by

T(X) _titImod X(m)e',(in). 4

(3.7)

In this curious. expression, the factors correspond to the multiplicative group of reduced residues mod q and the additive group of all residues modq. The absolute value of r(X) is found below. Gauss (preceding Dirichlet) considered only characters X for which X(m) takes the values + 1 and zero only. In this case T2 is real, but it is still not easy to find the argument of -r(X).

INTRODUCTORY RESULTS

12

1.3

We shall use T(X) to remove characters from a summation. With X(m) denoting the complex conjugate of X(m) (the inverse of X in the group of characters), eqn (3.7) gives .

(3.8)

T(X)X(m) _amodq I X(a)eq(am)

whenever (m, q) = 1. If T(X) is non-zero, we can use (3.8) to change a summation over X(m) to a summation over the exponential maps eq(am), which are easier to manipulate. A defect in eqn (3.8) is the condition (m, q) = 1. If, however, X is proper mod q eqn (3.8) holds for all integers m. We must show that the sum on the right-hand side of (3.8) is zero when q = tr, (3.9) In = tea, t > 1.

In this case,

amodq

X(a)eq(am) = I X(a)er(an).

(3.10)

amodq

Since X is not induced by any character mod r, there is an integer b with (b, q) = 1, b = 1 (modr), but X(b) zA 1. Our standard proof now applies. Multiplication by X(b) permutes the residue classes in the sum on the left of eqn (3.10), but multiplies the value of the sum by a constant that is not unity. The sums in (3.10) and (3.8) are thus zero if X is proper mod q and (m, q) > 1. For eqn (3.8) to be of use we must be sure that T(X) is non-zero. When X is proper mod q there is an elegant demonstration. For each m mod q,

I

IX(rn)T(X)I2amodq = I bmodq

Hence

X(a)X(b)eq(am-brn).

E IX(m)I2IT(X)(2aniodq = I bmodq I X(a)X(b) I mmodq

mmodq

eq(am-bm).

(3.11)

(3.12)

The inner sum is zero by eqn (3.1), unless a = b, when it is q, so that q I IX(a)12. JT(X)12 1 1 X(mn)12 =amodq mmodq

(3.13)

Since X is proper mod q if and only if its inverse X is proper mod q, we

deduce that

IT(X)12

= q for X proper mod q.

(3.14)

Equation (3.8) thus applies when X is proper mod q, the factor T(X) being non-zero. We conclude with the case when X mod q has conductor f, and q = fg

where g > 1. If the lowest common multiple h = [f, g] of f and g is not q itself, X is not merely induced by some character X2 mod h, but is actually equal to X21 since any integer prime to h is already prime to

fg = q. Replacement of m in the sum in eqn (3.7) by m+h permutes the residue classes mod q, but multiplies T(X) by eq(h), which is not unity. In this case therefore T(X) is zero.

CHARACTERS

1.3

13

When (f, g) = 1 we invoke the Chinese remainder theorem. By eqn (1.2) there are integers it, v with

fu+gv = 1.

(3.15)

Residue classes a (modfg) correspond to pairs of classes b (rood f ), c (mod g) according to the relation (3.16) a - cfu-]-bgv (modfg). In (3.16), a (modfg) is a reduced class if and only if both b (mod f) and c (mod g) are reduced, and thus T(X) _ J* X(a)eq(a) amodq Y_*

J* X(cfu+bgv)ea(cu)e1(bv)

bmodf cmoda b

momo

J*t X(b)ef(bv) c 1 as ea(cu)

= VV)T(X1)r-a(2L),

(3.17)

where X1 mod f is the character inducing X mod q, and ca(u) is Ramanujan's sum, defined in eqn (1.11). We now proceed to compute Ramanujan's sum. By eqns (1.11) and (1.27),

ca(u) _amoda I ea(au) (l a p(d ) d+a

d

p(d)bmYald ealca(bu),

(3.18)

where we have written a = bd. From eqn (3.1) the inner sum is zero unless it is a multiple of g/d. Writing h = g/d, we have ca(u) _ I hµ(g/h). (3.19) hh lu

It is possible to continue and to express ca(u) in terms of Euler's cp function. In our application, eqn (3.15) ensures that (g, u) = 1 and thus that ca(u) is µ(g), which is itself zero if g has a repeated prime factor. Ramanujan's sum with (g, u) = 1 can be regarded as Gauss's sum for the trivial character Xo modg, for which Xo(m) = 1 whenever (g, n2) = 1.

In this chapter, we have shown that T(X) is zero unless g = q/f is composed solely of those primes whose squares do not divide q, in which case f. (3.20) IT(X) I2 =

4 POLYA'S THEOREM `About as big as Piglet,' he said to himself sadly. 'My favourite size. Well, well.' I. 85

FROIT eqn (3.2) we see that the sum function

X(x) = mI<_X X(m)

(4.1)

is bounded when X is a non-trivial character mod q. Since the sum over any q consecutive integers is zero, the absolute value of X (x) can be at most -ffIq. P61ya (1918) proved the following sharper result. THEOREM. Let X be a non-trivial character inodq with conductor f. Then

I

tn5.

X(m)I < (1r-1+o(1))f d(q/f)logf,

(4.2)

where the term o (1) is to be interpreted as f -> cc.

Polya's theorem was discovered independently by I. M. Vinogradov (1955, chapter 3, example 12), with a different constant in the upper bound. Later proofs have been given by Linnik and Renyi (1947) and by Knapowski in an unpublished manuscript. We shall follow Polya's argument, as it is the most precise and can easily be adapted to show that the sum in (4.2) is frequently > f . First we introduce some notation. If a is a real number, we write [a] for the largest integer not exceeding a, and IIaII for the distance from a to the nearest integer, so that (4.3) [a] = maxm, m_
(4.4) IIaII = minim-aI, where the maximum in (3.3) and the minimum in (3.4) are over all integers mn. We now state a lemma.

LEMMA. The Fourier series

H (a )

_

11

2A _ - Go

va#0

e(ma)

(4.5)

POLYA'S THEOREM

1.4

15

if a is not an integer, if a is an integer, and the partial sums satisfy the relation converges to

0

nl

1 e ?a

1

H

-171

1

2Irlnb

2ir111IIaII

?fa#0

when a is not an integer.

Proof. We prove (4.7). Since H(a) has period 1 and H(-a) = -H(a), we suppose that 0 < a < 2. Now f e(mt) dt,

(4.8)

1

1 e(mt) dt 1

and thus

(2aim)

-ill

-117

Vx#0 U

J

-

e(111t+ 2t)-e(-Mt-fit) dt e(2t)-e(-1t)

(' sin(2111+1)7rt J Sill in

dt '

which by one of the mean-value theorems for integrals does not exceed in modulus

f sin(2M-F1)7rt dt

J

Sill Ira

2

1

1

111+1)or2irM <

(4.10)

for some R between a and 1. This has proved (4.7), which gives (4.6) if a is not an integer; if a is an integer every term in H(a) is zero. We now return to the stun in eqn (4.1). Since X is non-trivial, we have r'R +4

I X(m) = 0,

(4.11)

2q+1

and so we can replace the sum 1I (x) by a sum of the same form with

0 < x < q. If now x > -1q, we can replace m by -q+m and obtain a sum with 0 < x < 2q, multiplied by -X(-1). Thus we can suppose that 0 < x < jq in (4.1). We now use H(a) to construct G(a), where 1

G(a)

-

0

if 0
ifa=florax/q, if x/q
(4.12)

INTRODUCTORY RESULTS

16

1.4

using the equation

G(ot) = x/q+H(a-x/q)-H(a).

(4.13)

We now have e

X(1)+X(2)+...+zX(x)

m=1

X(m)G(rn/q).

(4.14)

When we use eqn (4.7) to truncate the sums for H(a-x/q) and H(a) in (4.14), the total error in modulus is at most i 2 q

(2ir111jjm/qIj)-1 < 4

2 ?n=1

n=1

2Trm2M

2ir-1111-1q log q.

(4.15)

We now have finite sums to manipulate. Writing M

G(a) _ I a(n)e(rna),

(4.16)

we have to consider X(r) 1

eq(mr).

(4.17)

Supposing first that X is proper mod q, we have from eqn (3.8) az

DI

Q

Y_ a(m) Y X(r)eq(inr) = I a(m)X(m)T(X). -Al . 2-1 -M

(4.18)

Since X is non-trivial a(0)X(0) is zero, and by eqns (4.5) and (4.13), for

m ' 0'

I a(m) I < hrrn I -1.

(4.19)

The modulus of the expression in (4.18) is now V


u na=1

< 2a-1fllogM+0(1)).

(4.20)

M = q1+s,

(4.21)

9r-1 q1 log q(1 + 0(8) ),

(4.22)

We choose

so that (4.20) is

and after (4.15) the tails of the series give < (irM) -1 q log q = o (ql log q).

(4.23)

Finally the omitted term 2X(x) is 0(1), and we have Polya's theorem when X is proper mod q.

P6LYA'S THEOREM

1.4

17

If X is induced by Xi proper mod f, where f < q, we could complete the proof similarly, but it is easier to deduce this case.

I X(m) _ In<x I X1(m) dQ .1 µ(d)

na<x

d1n

(d, )=1 (et,y) =1

µ(d)X1(d) Iid X1(,ni) m

f}logf{ir-i+o(1)},

(4.24)

(d)t1 ,t since Xi is proper mod f. The number of terms in the sum is at mos d(q/f ), and we have completed the proof of Polya's theorem. As a simple corollary we prove that for each prime p > 2 there is an I

integer m with

m
(4.25)

for which the congruence (4.26) u2 - m (modp) has no solution. Since 1, 4, 9,..., 1(p-1)2 are distinct unodp, there are "I(p-1) reduced residue classes that are congruent to squares modp, and these form a subgroup of index 2. There is therefore a character

X modp with X(m) = 1 when m is congruent to the square of a reduced

residue and -1 when (m, q) = 1 but m is not congruent to a square. We choose (4.27) x > p1logp in (4.2). Not all terms in the sum (4.1) can be non-negative if p is sufficiently large, and so there is an m < p! log p with X(m) = -1. The exponent in (4.25) can be improved, but the conjecture that the asymptotic inequality (4.28) In 0 holds in place of (4.25) has not yet been proved or confounded.

5

DIRICHLET SERIES 'Well,' said Owl, 'the customary procedure in such cases is as follows.' 'What does CSrustimoney Proseedcalce mean?' said Pooh. 'For I am a Bear of Very Little Brain, and long words bother me.'

'It means the Thing to Do.'

1.48

A Dirichlet series is an analytic function of the complex variable s = a-i-it defined by a series Co

a(rn)m-s,

f (s)

(5.1)

ma=1

or a generalization thereof. All the Dirichlet series that we need are special cases of (5.1). If eqn (5.1) converges at so = ao+ito, then Ia(m)m-so j is bounded. This simple observation is the basis for the theory of convergence of Dirichlet series. By partial summation (5.1) converges whenever a > ao and converges absolutely when a > ao+1. The convergence is uniform in a half-plane a > a, provided a > ao+1. We see that the region of definition off (s) is a half-plane bounded to the left by some vertical line; this line is called the abscissa of convergence.

If

A(x) _

a(m)

w<x

(5.2)

is the sum function of the coefficients in (5.1), then 00

f(s) = f sx-s-1A(x) dx.

(5.3)

1

Formula (5.3) can be inverted: from f (s) we can recover the sum function

A(x) of the coefficients. Let a and it be positive real numbers. Then 01+100

f usds =

tai J

0

s

if 0 < u < 1, if it = 1 ,

(5 4) .

if it > 1. For it 1 we consider the integral from a-iT1 to a+iT2. When it > 1, this is equal to the integral round the three remaining sides of the «-goo

1

DIRICHLET SERIES

1.5

19

rectangle whose other corners are R/logu+iT2, R/loges-iT1. The modulus of the integral in eqn (5.4) is thus eG-a-e-R R-1e-R 2G-a-e-R 2i

(5.5)

+ 27TTilogu2,r121ou' +

a number which tends to zero as B, Ti, and T2 tend to +oo. When u < 1, B/log it is negative, and we must add the residue from the pole

of s-1 at s = 0; this gives unity. Finally when it = 1 we define the value of the integral (5.4) to be the limit of the integral from a-iT to a+iT when T -+ oo. This reduces to an inverse tangent integral.

R/ log u

a

(7

FIG. 1

If f (s) defined by eqn (5.1) converges uniformly in t on the line a = a,

then for x > 0 term-by-term integration gives

«+i

f

xss -If

(s) ds

(5.6)

a-coo

where the last term occurs only if x is an integer. There are many integral transforms from Dirichlet series to their coefficient sums, all proved by the same method. The simplest one after (5.6) itself is «+iOO

f

21T1

xsf(s) 8(8 +1)

ds =

7A<'c

(1-7JZla(m). I.

(5.7)

iao

A special class of Dirichlet series occurs naturally in number theory. The functions are known as L -functions after Dirichlet's functions 00 L(s, X) =m=1 I

X(i1)m-s,

(5.8)

where X is a Dirichlet's character to some modulus q, or as zeta functions

after Riemann's function m=1

(5.9)

20

INTRODUCTORY RESULTS

1.5

which is the special case of Dirichlet's definition when q = 1 and X is trivial. L-functions are defined by two properties. First, the coefficients a(m) are multiplicative, so that Euler's product identity ans)

_

(1+

(5.10)

holds in a half-plane a > a in which one side of eqn (5.10) converges absolutely. If the product in (5.10) converges, f (s) can be zero only when one of the factors on the right-hand side of (5.10) is zero. The convergence of the left-hand side of (5.10) alone does not imply that of the product; L(s, X) with X non-trivial has a series (5.8) converging for a > 0, but the function itself has zeros in a > 2, preventing the product from converging in 0 < a < J. The second defining property is that f(8) should have a functional equation f(s)G(s) = f*(r-s)G*(r-s), (5.11)

where r is a positive integer, G(s) is essentially a product of gamma functions, and the operation * has (f*)* =,f and (G*)* = G. As an example, in the functional equation for L(s, X) in Chapter 11, L*(s, X) is L(s, X). An important conjecture about L-functions is the Biemann hypothesis that if f (s) satisfies eqns (5.10) and (5.11) then all zeros of f (s) G(s) have real part Jr. The truth or falsity of this hypothesis is not settled for any L-function. Two generalizations that are often called zeta functions are co

(m } 8)-S,

(5.12)

m.=1

where 8 is a fixed real number, and co

1 r(M)M-8,

m=1

(5.13)

where r(m) is the number of representations of m by a positive definite

quadratic form. Except in special cases these fail to have a product formula of the form (5.10), and not all of their zeros lie on the appropriate line. Some authors even use `zeta function' as a synonym for `Dirichlet series'. In Chapter 11 we shall obtain analytic continuations of c(s) and other L-functions over the whole plane. Since the sum function X (x) formed

DIRICHLET SERIES

1.5

21

with a non-trivial character X is bounded, by partial summation (5.8) con-

verges for a > 0 except when X is trivial. Similarly, the function

I 00

(-1)m-1m-3

m=1

(5.14)

= (1-21--%(s)

converges for a > 0 and provides an analytic continuation for c(s). When we make s - 1 in (5,14), we see that c(s) has a pole of residue 1 at s = 1. When we put f (s) = c(s) in (5.6), the integrand has a simple pole at s = 1 with residue x. The value of the right-hand side of eqn (5.6) is between x-1 and x. If we deform the contour in (5.6) so that it passes to the left of the pole, the residue makes the main contribution, and the contour integral left over is bounded. Let

O(x) _ I A(m)

(5.15)

ma<x

and

117(x) _

(5.16) rn <x

and of 1/c(s) (we shall prove

which are the coefficient sums of

this below). The function also has a pole of residue 1 at s = 1, but 1/c(s) does not. If the corresponding contour integrals were negligible, we should have

OM = x+o (x),

(5.17)

M(x) = o (x).

(5.18)

These are forms of the prime-number theorem, which we shall prove in

Part II. Writing m = fg, we have 00

I qn-8 fins I a(f)b(mlf) 7)L=1

f=1

a(f )f -3)0=1 ( b(g)g-).

(5.19)

If b(g) = 1 and a(f) = µ(f) for each pair of integers f, g, Co

(s) m=1 µ(7n)m-9 = m=1 n-3 1

1

(5.20)

flm

from eqn (1.27). Since eqns (5.3) and (5.6) imply that expansions in Dirichlet series are unique, we have shown that the series on the left-hand side of (5.20) represents 1/c(s) wherever it converges. Similarly, using eqns (5.19) and (1.32) we can check that -t'($)/C(s) has a Dirichlet series with coefficients A(rn). For fifty years (1898-1948) the only proofs known of eqns (5.17) and (5.18) used contour integration and other complex-variable techniques.

In 1948, Selberg and Erdos gave a real proof of (5.18) (see Hardy and

INTRODUCTORY RESULTS

22

1.5

Wright (1960), Chapter 22). The real-variable approach is not so well understood, and the strongest forms of (5.17) and (5.18) (those in which the error term is smallest) have been obtained by analytic methods. The form (16.22) in which we shall prove (5.17) is a little stronger than the best so far obtained by Selberg's method. Apart from the analytic arguments, study of log(N!) suggests the form (5.17) as a conjecture. By eqn (1.32), log(N!) I log ??z = A(d) m
de-
= I O(N/e), e
(5.21)

where we have written m = de in the first sum. On the other hand, by expression (2.5),

I logm = ?n
m6N T fl -1

N-r-1 -ivy-I-o(logIV)

r5N

r

= NlogN-N+O(logN),

(5.22)

which agrees with the result of substituting (5.17) into (5.21). In this way, Gauss was led to conjecture that x

1 = {1+o (1)} f (log t)-1 dt, P<-x

(5.23)

2

which is another form of the prime-number theorem. By consideration of the binomial coefficient 2,v0N, it can be shown that 0(2N)-O(N) lies between bounded multiples of N (Hardy and Wright, 1960); but there

are too many terms in the sum (5.21) to allow (5.17) to be deduced from (5.22).

6

SCHINZEL'S HYPOTHESIS And all the good things which an animal likes Have the wrong sort of swallow or too many spikes.

II. 30

MANY problems in prime-number theory follow a similar pattern. Various constraints are laid on a set of integer unknowns, and we ask whether the integer unknowns can all be prime simultaneously, and whether this happens infinitely often. Many of these problems are subsumed under Schinzel's hypothesis: if fl(xl,..., x1),..., f ..(xl....... ,y) are polynomials (with integer coefficients) irreducible over the integers, and

there is no prime p for which II fi = 0 (mod p) for all sets xl,..., x, of residues modp, then there are infinitely many sets x1,..., x, of integers for which the absolute values of fl,..., are all prime. There is a conjectured asymptotic formula for the number of sets xl,..., x,, with 0 < xi < N for each i with each of fl,..., frn prime, and some bold authors have conjectured that the asymptotic formula can

be stated with an error term smaller than the main term by a factor Nl-E for each a greater than zero. Thus the simplest case is one polynomial, f (x) = x, and the conjecture now states that 7r(N), the number of primes up to N, satisfies the relation N

or(N) = f logx+ 0(N1+%

(6.1)

2

a very strong form of the prime-number theorem (5.23). The accuracy of (6.1) seems unattainable. We shall prove later that the hypothesis is true for one linear polynomial f (x) = qx--a; this is the prime-number theorem for arithmetical progressions, but the error term in the asymptotic formula will only be shown to be slightly smaller than the leading term. The next simplest case concerns two linear polynomials, fl(x) = x,

f2(x) = x-2. Here the conjectured formula is N

21T {1-(p-1)-2} f (log t)-2 dt +error term, P> 3 863618X

2

(6.2)

INTRODUCTORY RESULTS

24

1.6

We shall now describe how to write down the conjectured asymptotic

formulae. Let

S(a) _

e(pa);

ps

(6.3)

such an expression is called an exponential sum or a trigonometric sun. By the fundamental relation 1

f e(ma) da

1

=

if m = 0,

t0 if m

0,

(6.4)

we see that the number of primes p < N for which p-2 is also prime is 1

f 8((x)8(-a)e(-2a) doz.

(6.5)

0

We cannot, of course, work out this integral, but we can suggest a plausible value for it. Writing 1, 7r(N; q, b) _ p= (modq)

&

we have

S(a/q) _

eq(ab)ir(N; q, b).

(6.6)

(6.7)

b mod q

Now the sum in eqn (6.6) is 0 or 1 if b has a common factor with q. If we make the approximation that the primes are divided equally between the p(q) residue classes b with (b, q) = 1, the expression in (6.7) is {p(q)}-1,T(N) J* eq(ab) = {,p(q)}-1ir(N)cq(a), b modq

(6.8)

where cq(a) is Ramanujan's sum (1.11). If (a, q) = 1 the Ramanujan's sum is just µ(q), from eqn (3.19). This argument suggests that S((X) has a `spike' at a/q of height proportional to F.c(q)/p(q), that is, that IS(a) I has a local maximum close to a/q. Now the area under the graph of I S(a)12 near 0 (which certainly is the site of a spike) may plausibly be written N-17T2(N) +error term, (6.9)

and this can be proved to be true. If we assume further that all the spikes at rational points are the same shape, the spikes at rational points a/q contribute N--11T2(N) I p.2(q)(pp(q))-2 J* eq(2a)+error term, amodq

q

(6.10)

and the sum over q in (6.10) converges to

2 J {1-(p-1)-2}, p>2

which gives the main term in eqn (6.2).

(6.11)

1.6

SCHINZEL'S HYPOTHESIS

25

For small q the argument above can be made rigorous; but then part of the range of integration in (6.5) does not support spikes. Away from a spike we cannot estimate S(a) except by replacing it by its absolute 1

value; and the spikes with small q contribute very little to f IS(a)12 da. 0

In the integral of IS(a)13 the spikes do dominate, and by this method I. M. Vinogradov was able to prove that every large odd number is the sum of three primes. The approach to Schinzel's hypothesis through exponential sums does lead to an upper bound for the number of sets x1,..., x,, of integers not exceeding N for which fl,..., are all prime. To explain the method

we shall take n = 1, so we are considering integers x in the range 1 < x < N for which fl(x),..., f,,,(x) are all prime. We now work modulo a prime p. Apart from the finite number of x for which one of f1(x),..., f,,,(x) is p, x must be such that none of f1(x),..., 0 (modp). This

means that x must be confined to certain residue classes modp. We therefore divide the residue classes modp into a set H(p) off (p) forbidden

classes and a set K(p) of g(p) = p-f(p) permitted classes; lm modp is forbidden if and only if one of the polynomials ff(h) is a multiple of p. If x falls into a forbidden class for any prime p smaller than each of the fa(x), then one at least of the fi(x) cannot be prime. The values of x that make f1(x),..., f,, (x) primes greater than some bound Q form a sifted sequence, in the following sense. The increasing sequence . 4" of positive integers n1, n2,... is sifted by the primes p < Q

if for each prime p < Q there is a set H(p) (possibly empty) of f(p) residue classes modp into which no member of .N' falls. We shall show

in Chapter 8 that, if .N' satisfies the above condition, the number of members of ." in any interval of N consecutive integers is N q'Q

where

+error term,

(6.12)

µ2(q)f(q)lg(q)

f (q) = q IT f (p)/p,

(6.13)

pig

g(q) = q f {1 f (p)/p}.

(6.14)

pig

We shall work out examples of this upper bound in Chapter 8; in each case the leading term is a multiple of the leading term in the conjectured formula. Upper bounds of the right order of magnitude were first found by Viggo Brun using combinatorial arguments. Rosser used Brun's method

INTRODUCTORY RESULTS

26

1.6

to obtain expression (6.12), which was found in a different way by Selberg.

An outline of Selberg's method follows. It rests on the construction of kin exponential sum T(a) with the same spikes at rational points as S(a) _ I e(nza)

(6.15)

9bKN

is conjectured to have. Let K(q) be the set of g(q) residue classes that are in K(p) (permitted classes) for each prime factor p of q. At a/q we expect a spike of height proportional to

K(a, q) =keK(q) I eq(ak)lg(q).

(6.16)

If p is a repeated prime factor of q, then the classes lc+glp are also in K(q), and since replacement of le in eqn (6.16) by k+q/p multiplies the right-hand side by e2,(a), which is not unity, K(a, q) is zero if q has any repeated prime factor. Now the simplest exponential sum is

F(a) _

N m=1

e(9na),

(6.17)

which has a spike at a = 0, since JF(a) = Isin 1rNa/sin 1ral.

(6.18)

We therefore compare S(a) with

T(a) = I J* x(a, q)F(e -a/q). q6Q amodq

(6.19)

The coefficient of e(ma) in T(ae) is

rJ

*

1

q Q g(q)

eq(alc-am) =

1

q

q) amodq

k

g(g) k K(4)

(6.20)

using the definition (1.11) of Ralnanujan's sum. We know the sum over 7c must be zero if q has a repeated prime factor; if q is square-free then

I

cq(k-mn) =

dµ(dq)

ICEK(q)

kZr( q)

al

dl (k 4m)

q g(q)

d g(d) II

µ(q)g(q)µ(g1)f(g1)lg(g1),

(6.21)

where q1 is the largest factor of q for which rn E K(q1). We write q = q1 q2, so that if mz E H(cl) then d divides q2. The expression in (6.21) becomes

µ(g2)g(g2)f(q)lf(g2)

(6.22)

SCHINZEL'S HYPOTHESIS

1.6

f(d)

p(d)d

Now djq2

27

p(g2)J(g2)

(6.23)

f(q2)

and so the coefficient of e(ma) in eqn (6,19) is µ(d)d

=a

)

11 2(q)f(q)

f(d) q°R
(6.24)

g(q)

Selberg's argument proceeds as follows. If m is in .N', then m e H(d) only when d = 1, so that by eqn (6.4) µ2(q)f(q)

f1 S(a)T(-a) M
0

(6.25)

J(q)

We obtain (6.12) by Cauchy's inequality, 1

1

1

f S(a)T(-a) dale < f IS(a)I2 da f IT(fl)I2 dp, 0

(6.26)

0

0

noting that, from eqn (6.4), 1

1

f IS((X)I2 dot

and

f I T(a) I2 da 0

=

f S(a)S(-a) dot !k2(q)f (q) -{-

=N q
J(q)

=

1,

(6.27)

N< N

error term.

(6.28)

We can prove eqn (6.28) either directly from (6.19) or by writing I T (a) I2

as T(a)fl-a) and using (6.4) again and the expression (6.24) for the coefficients. Selberg used the second method, which can be expressed entirely in terms of the manipulation of the coefficients (6.24), with the integrations (6.25) and (6.28) occurring only implicitly. The first proof uses eqn (6.18) to show that the spikes of T(a) provide the main contribution to the integral in (6.28), and the observation that J* IK(a,q)I2 = µ2(q)f(q)/J(q). a mod q

(6.29)

The usual account of Selberg's method in terms of coefficient manipu-

lations can be found in Halberstam and Roth (1966, Chapter 4). The sketch above is presented in terms of exponential sums for comparison with the `large sieve' of the next two chapters.

7

THE LARGE SIEVE But whatever his weight in pounds, shillings and ounces, He always seems bigger because of his bounces. 11. 30

W r,- have seen that the behaviour of a given sequence of integers considered mod q is reflected in the behaviour of the stuns S(a/q) where

S(a) _N a e(ma), L

(7.1)

1

in which is 1 if m is in the given sequence, and 0 if mn is not. An upper bound for the sum J* I S(a/q) 1 2 (7.2)

I

amodq

(or for some related expression) is called a large sieve for residue classes.

Other large sieves will appear in Chapter 18. In this chapter we prove what is probably the simplest of the upper bounds for the sum (7.2), and in the next chapter we shall use it to prove an upper bound of the form

(6.12) for the number of elements of a sifted sequence in a bounded interval. The proof does not require that am in eqn (7.1) takes only the values 0 or 1, and it treats (7.2) as a special case of the sum RR

G I S(xr)I2

=1

(7.3)

where 0 < x1 < x2 < ... < xR < 1. Since a/q occurs in the sum (7.2) only if it is in its lowest terms, the points x,, are distinct in our proposed application. We write 8 = min{x2-xl,x3-X2, ...,x1+1-xR} (7.4) and suppose that 8 > 0. Before proving an upper bound for (7.3), we consider what form it might possibly take. Certainly there will exist sequences of coefficients a,,, and points x,, for which R

G t=1

(1'

8-1 J I S(a)12 da 0

= 8-1 G lamI2. CN'

1

(7.5)

THE LARGE SIEVE

1.7

29

On the other hand, any one term in the sum (7.3) may be as large as Ia,,, 1)2, and if all the are equal in modulus this is

N

N

(7.6)

la,I2.

1

An optimistic conjecture is that the inequality N

R

r=1

IS(xr)I2 < (N+8-1)

(7.7)

la,,,I2 1

always holds. Surprisingly, the right-hand side of (7.7) has the correct order of magnitude: we shall prove that N

R

(7.8)

IS(zr)I2 < (N+28-1-V3+0(1)) I la,,,I2; r=1 1 this result is due essentially to Bombieri (1972).

To prove the relation (7.8) we use the language of N-dimensional vectors over the complex numbers. The inner product (g, h) of two vectors g = (y1,..., gN) and h = (h1,..., hN) is given by N g"

(g, h) _ and the norm IIgII by

(7.9)

(7.10)

11911 = ((g,g))fr.

We can now state a fundamental lemma. LEMMA. Let u, f(1),.,., f(R) be N-dimensional vectors, and c1,..., cR be any complex coefficients. Then r=1

cr(u,f(r))I < IIuIi(

IcrI2)( max

-1,...,R s=1

1

I(f(''),f(s))I)}.

(7.11)

Proof. The left-hand side of (7.11) is

(u,

(7.12)

crf('))I < IIuIIXIIY ',.f(''>II.

The square of the second factor on the right-hand side of (7.12) is R

R

R

I I grgs(f(' ,f(s)) < e r=1 r=1 s=1

_

R

r=1

R

f(s))I

s=1

R IC,,I2 1 I(f(r),f(R))l,

s=1

(7.13)

from which the result follows. In applying the lemma we choose Cr so that each summand on the left of (7.11) is real and positive. In Chapter 27 we shall apply the lemma

INTRODUCTORY RESULTS

30

1.7

with c, of unit modulus to obtain Halasz's method for estimating the number of times S(a) is large. To prove (7.8) we take c, given by c,. _ (f,", Fs))

(7.14)

and deduce the corollary. COROLLARY. We have R

R

I(u,f(''))12 < IIuII2 max r=1 15r`R s=1

We write

(7.15)

I(f(,'),f(s))I.

H!

S(a) = e(111a+a)

- ill

be(mna),

(7.16)

where 2111 = N or N-1, whichever is even. We choose f(1.)

-

(7.17)

k»,e(-9nx,.),

(7.18)

klr are given by

where

k

for 111

< Imi < 112-+ ;

k,,, = 0 for I mn I > 111+L

(7.19)

1

here L is a parameter which we shall choose below. We now have nW

I(u,P'))I = I

ill

Moreover

IIuII2 = G Ib,ni2/k2 and

_

IS(x,.)I.

(7.20)

Iam12

(7.21)

N

(V''), f(s)) = Ic(xs-x,.),

where

K(oz) _

(7.22)

r1I+L

-t1L-L

k2, e(mna)

sin2(111+ L)7ra-sin21117ra

(7.23)

L si1127TO,

Hence we have R

I < 2111+L+2

s=1

L-1 cosec27rm8 1 00

2111+ L +

2L-11T2

7n-28-2+O(L-1)

2M+L+(3L82)-1+O(L-1).

(7.24)

1.7

THE LARGE SIEVE

31

When we choose L to be an integer close to (8 \/3)-1, (7.24) becomes 2111-} 328-1d3+O(1).

(7.25)

We substitute (7.20), (7.21), and (7.25) into (7.15) to obtain (7.8). Our inequality (7.8) represents an improvement of an inequality of Roth (1965), which has led to much recent work. The best upper bounds known for the sum (7.3) at the time of writing are 2-\/3

N

_\T)

3

(7.26) 1

8-1(1+270N383)

N (7.27) 1

and

2max(N,5-1)

Iv

1a12.

(7.28)

1

Of these, (7.26) is the result of this chapter, appropriate when N > 18-11/3, and (7.27) and (7.28) are results of Bombieri and Davenport (1969, 1968).

(7.27) is appropriate when 8-1 > 3(10)'N, and (7.28) for the intermediate range.

Note added in proof. H. Montgomery and R. C. Vaughan have now proved the conjecture (7.7). This supersedes (7.26) and (7.28) but not (7.27).

8

THE UPPER-BOUND SIEVE `It's a comfortable sort of thing to have', said Christopher Robin, folding up the paper and putting it into his pocket. IT. 170

IN this chapter we obtain the upper bound (6.12) as an application of the large sieve. The notation is that of Chapter 6. A,' is a sequence of positive integers, and for each q < Q there are sets H(q) and K(q) of residue classes mod q. The f (q) classes of H(q) are precisely those that are not congruent to any member of .N' mod p for any prime p dividing q,

so that, if h is in a class of H(q) and n e .N',

(n-h, q) = 1.

(8.1)

The g(q) classes of K(q) are those that for each p dividing q are congruent

modp to some member of Al; their union contains all members of the sequence X. We work with the exponential sum of eqn (6.15):

S(a) _ I e(n2a),

(8.2)

91j
where the sum is over members nl, n2,... of the sequence .4". If h is a class of H(q),

J* S(a/q)eq(-ah) _ I cq(nz-h) itj
amodq

= µ(q)111,

where M is the number of members of.A", and we have used eqn (3.19) in

the special case (8.1). Hence pp(q)f(q)111 = .J*

(8.4)

I S(a/q)eq(-ah).

amodq he7l(q)

Cauchy's inequality now gives µ2(q)f 2(q)1122 < (amodq I S(a/q) I2) (a Idq I

I

li

q(-ah) (q)

12).

(8.5)

THE UPPER-BOUND SIEVE

1.8

33

The second sum over a on the right-hand side of (8.5) can be rearranged as follows. hI(q)eq(-ah)

a Jdq

a

I

a(q)

H(q)eq(ag)

hI

(q)

gc-

(q)eq(g-h)

dj

q,,)dµ(gId)

_ Ydp.(q/d)f2(q)lf(d). dlq

(8.6)

In eqn (8.6) we have a sum that is a Mbbius inverse (in the sense of (1.27)) of that in N2(r)g(r) Inn

f (r)

m f (m)'

an equation that expresses the fact that for a prime modulus p, every residue class is either in H(p) or in K(p). The terms involving d in (8.6) therefore come to µ2(q)g(q)/f (q), and we can put (8.5) into the form !J_ 2(q)f (q).312

g(q)

<

*

(8.8)

q amodq JS(a)12.

We apply the large sieve (7.8) with the rationals alq with q < Q and (a, q) = 1 as the points x1,..., XR, so that

8 = (Q(Q-1))-1

(8.9)

in eqn (7.4). The upper bound (7.8) now gives

I

'j* I S(a/q)12 <

(N+0(Q2))111.

q5Q amodq

(8.10)

Combining (8.8) and (8.10), we have 1112 1 t42(q)f(q)lg(q) < (N+O(Q2))M, or

112 <

N+a(Q2) 1 µ2(q)f(q)lg(q)'

(8.11)

(8.12)

q
which is the relation (6.12) with an explicit error term. We have stated

our result for the interval 1 < n < N of the sequence .it, but (8.12) holds as an upper bound for the number of members of .N' in any interval

L+1 < n < L+N; we have to consider merely the integers n- L, where n is in .N', themselves a sifted sequence with the same values of f(q) and g(q) but the sets H(q) and K(q) translated by L. The deduction of (8.12) from (7.8) is due to Montgomery (1968). His argument differs from that given here; there are several alternative proofs.

INTRODUCTORY RESULTS

34

1.8

Two worked examples follow. First we consider the perfect squares not exceeding N. These are a sifted sequence: the set H(p) contains the residue classes modp that do not contain squares, and thus g(2) = 2 f(2) _ 0,

g(p) = 2(p+1) for p >

f(p) = 2(p-1),

3).

(8.13)

It is not difficult to show that q

Qµ2(q)f(q)lg(q) = (c+o(1))Q,

(8.14)

where c is a constant, as Q -a oo. Choosing Q = N1, we have shown that the number of perfect squares not exceeding N is O(N1). This is very encouraging, since the sieve upper bound is `sharp', differing only by

a constant factor from the actual number of squares. It is surprising that we have not lost the correct order of magnitude in combining so many inequalities. Our second, less trivial example concerns the primes between Q and N; these form a sifted sequence with f (p) = 1,

g(p) = p- I =
µ2(q) f(q) _

and thus q5Q

(8.15)

µ2(g.)

2(p (q)

qQ

g(q)

_ (1+(1))logQ,

(8.16)

when worked out as in Chapter 2. We can obtain a lower bound for the sum in (8.16) with less effort, since µ2(q)

_

2(q) qSQ

q6Q `P(q)

q

1

Piq

1

p

_ I 1/m,

(8.17)

the sum being over all m whose prime factors are at most Q. All in < Q are included in this sum, and thus µ2(q) q
y(q)

>

> log Q

(8.18)

9m5Qin

for q > 1, by (2.5). When we choose Q a little smaller than N1 we see that the primes between 1 and N number

< Q+(N+O(Q2))/logQ < (2+o(1))N/logN.

(8.19)

The right-hand side of (8.19) is just double the true value (5.23). More-

over, (8.19) is also an upper bound for the number of primes in any interval of length N.

1,8

THE UPPER-BOUND SIEVE

35

We derived the inequality (8.12) from Cauchy's inequality; the difference between the two sides of the inequality (8.12) is a measure of how closely the values of S(a/q) are proportional to those of (J(q))-1 hEI e4(ak),

(8.20)

and this in its turn measures how evenly .A" is distributed among the g(q) residue classes mod q into which it is allowed to fall. We could add an explicit term on the right of (8.8) to measure the unevenness (what statisticians might term a variance). The inequality (7.8) gives a strong

upper bound for this variance as well as for the main term. When we use (7.8) to prove Bombieri's theorem, it is the variance bound that is important, not the bound for the main term.

9

FRANEL'S THEOREM `It's just a thing you discover', said Christopher Robin carelessly, not being quite sure himself. 1. 109

THE Fare y sequence of order Q consists of the fractions a/q in their lowest

terms (i.e. (a, q) = 1), with q < Q and 0 < a < q. We name them .f,. = a,./qr in increasing order, so thatfi = 1/Q, f2 = 1/(Q-1),..., fF = 1. For notational convenience we may refer to fF+,.; this is to be interpreted as 1-1- f,.. Here F is the number of terms in the Farey sequence, so that

F = I p(q) = 37r-2Q2+0(Q log Q) 45Q

(9.1)

from eqn (2.11). The properties of the Farey sequence are discussed by Hardy and Wright (1960, Chapter 3). We shall sketch a proof that (9.2) fr+i-f,. = (grgr+i)-1. Let us represent rational numbers a/q (not necessarily in their lowest terms) by points (a, q) of two-dimensional Euclidean space. Sincefr and .fr+1 are consecutive, the only integer points in the closed triangle with

vertices 0 (0, 0), Pr (ar, qr), and Pr+1 (ar+i, gr+1) are its three vertices. By

symmetry, the only integer points in the parallelogram OPr TP,.+1 are its vertices, where T is (ar+a,.+i, q,.+q,.+1). We can now cover the plane with the translations of this parallelogram in such a way that integer

points occur only at the vertices of parallelograms. It follows that OP,. TPr+1 has unit area, which is the assertion (9.2).

Before stating Franel's theorem we introduce some notation. For

0
1 we write (9.3) }fry«

so that E(a) is the excess number of Farey fractions in (0, a] beyond the

expected number F. Franel considered the sum F

r=1

E(fr) I2

(9.4)

FRANEL'S THEOREM

1.9

37

and showed that it is o (Q4) if and only if eqn (5.18) holds; and an upper bound QA for (9.4) with 3 < A < 4 is valid if and only if (9.5)

AX) I <X11-1. We can also connect M(x) with 1

f IE(a)12da.

(9.6)

0

In fact, the ratio of (9.4) and (9.6) lies between bounded multiples of F; but this fact requires proof, as the Riemann sum corresponding to the integral (9.6) and the points f1,..., fZ,, is

G, (fr+ifr) jE(fr) I2, A

(9.7)

and from eqn (9.2) we see that the difference fr+i fr varies from Q-1 almost down to Q-2. Franel (1924) produced a curious identity for the sum (9.4). We shall show in (9.20) that (9.4) is less than a bounded multiple of Franel's expression involving M(x), and deduce the `only if' clause of Franel's result by a method of Landau (1927, Vol. II, pp. 16977).

We use the function H((x) of eqns (4.6) and (4.6) :

a

H(a) = i

if a is an integer.

We have

- '-F H(a fr) _ (E(a) E(a)-4

if a is in the Farey sequence, if not,

(9.9)

for 0 < a < 1. To verify eqn (9.9), we observe that the left-hand side is aF plus a step function that is zero at a = 0 and has discontinuities - z on each side of the Farey points fr. Taking (4.7) in the form

H(a) + 2

e (ma) )

<

n= n#0

(9.10)

114'

we have from (9.9)

E(.fr) _ -tIP H(f,.-ft) Q9

t=1 N".

it

Qa

tx#0

e(mfr-v) ft) 2aim

1 +O\QZ11.f,-fl

ll

(9.11)

INTRODUCTORY RESULTS

38

1.9

By (9.2), the consecutive Farey fractions are at least Q-2 apart, and so the sum of the error terms in (9.11) is F`

Q2/t < log F < log Q, < Q-2G t

(9.12)

where we have used expressions (2.5) and (9.1). We can now replace

the term t = r (since H(O) = 0) and rearrange the first term on the right of (9.11) as

I

I (2vrim)-1e(infr-mfg) = 1 (21rim)-1 -Q'

t=1 m=-Q2

va#o

7n#0

J* e(mfr)eq(-am)

q
Q2

-

I

Q2

eq(-M).

(9.13)

gSOI

moo

We apply the large sieve (7.8) with (9.13) as the exponential sum and fi, , fF as the points. By eqn (9.2), 8-1 is here Q(Q-1). We have FI

2

Q2

/

G C4(-m)I < Q2 q<- Q

2

Q2

-W m-2I q<0 c m#o

< Q2 L.

m-21

m=1

(-m)l

Lr d

dim

G

I-t(1/d')I2,

q=-0(modd) (9.14)

where we have used eqn (3.19) for Ramanujan's sum. The coefficient of Q2 in (9.14) is now

< I d112(Q/d) I fill'(Q/f) f5Q

d-
I

m=0(modd) m=0(modf)

m-2,

(9.15)

where we have taken the sum over m from m = 1 to infinity, since the coefficient of rn-2 in (9.14) is non-negative. The inner sum in (9.15) is Ir2

1

fJ2 =

or2 (d, f)2

(9.10)

6 [d 6 d2/2 Let u(m) be the arithmetical function with I u(d) = m2.

(9.17)

dim

By Mobius's inversion (1.28), we can verify that 0 < u(t) < t2.

(9.18)

We can now write (9.15) as

21 dM(Q/d) I f111(Q/f)d-2f-2I u(t)< I u(t)( d
f
td tIf

t
d-1M(Q/d))2

d
d=o(urod1)

(9.19)

FRANEL'S THEOREM

1.9

39

Squaring (9.11) and substituting (9.12) and (9.19), we have 2=1

1M(Q/(l))2+F1og2Q

I E(fr)I2 < Q2 111(1) t5l)

r

d=-O(modt)

< Q2tI t2(

d-'M(Q/d))2+Q21og2Q.

d`B

(9.20)

d-0(mod t)

The leading term on the right of (9.20) is essentially the sum involving Mobius functions of Franel's identity. The inequality from (9.8) to 111(x) is less involved. Since, by (3.19), (9.21)

I* eq(a) = cq(1)

amodq

we have Now Since

from (3.1), we have

M(Q)

F

I e(fr). = r=1

(9.22)

Ie(fr)-e(r1F)I < 27rIfr-r/FI 27rF-1IE(fr)1.

F

I e(r/F) = 0

(9.23) (9.24)

r=1

F

IM(Q) I < 2,rE-1

I E(fr) I

E(fr) I2)1

<
IE(fr)12)1,

(9.25)

and so a bound for (9.8) of the form o (Q4) implies the prime-number theorem in the form (5.18).

863618

D

PART II

The Prime-Number Theorem

10

A MODULAR RELATION Winnie-the-Pooh read the two notices very carefully, first from left to right, and afterwards, in case he had missed some of it, from right to left. 1.46

IN this second part we study t(s) and L(s, X) as functions of the complex variables, and work towards the prime-number theorem. Our investigations are based on the functional equations for C(s) and for L(8' X). The first step is therefore to prove these. We need a lemma from the theory of Fourier series. LEMMA. (Poisson's summation formula.) Let k, l be integers. Let f (x) be a differentiable function of a real variable with

on [k, 1]. Then k

Y

If'(x)I
l

J f(x)e(mx) dx = zf(k)+f(k+1)+...+f(k+1-1)+If(lc+l) k

(10.1)

and moreover the partial sums satisfy the inequality M rk+l

V kf f (x)e(mx) dx- 2 f (k)-...-I f (1c+l)I < 1AM-1 log M. (10.2) Proof. Equation (10.1) is additive on intervals: its truth for [lc,r] and for [r, t] implies its truth for [k, t]; so we may suppose l = 1. By

A MODULAR RELATION

2.10

41

a change of variable we may suppose also k = 0. We use the function H(x) of eqn (4.4):

H(x) _ >' m=-00 m#0

e(mx)

-

-

x -2 if 0

<x<

1

if x = 0 or 1,

0

27r1m

(10.3)

and in particular if 0 < x < 1 we have (4.6) : M

H(z)+

(10.4)

I

(MIIxll)-1

27r19n

m#0

Hence of

-nr

e(4nx)

f I m-1 dx+

(2zrim)-1e(mx)I dx

I H(x)+

o

*100

1

2

f M-1x-1 dx

1

1j11r

C M-1 log M.

(10.5)

We now have nr,

1

f f (x)e(rnx) dx - stir

vn#0 0 MY

_ [f (X)

(2irim)-1e(mx)]1f f'(x) j (2irim)-1e(mx) dx. (10.6) 0 0

m#0

vn#o

The first term on the right of eqn (10.6) is zero, since the terms for m and -m cancel, and by (10.5) the second is

ff'(x)H(x) dx+O(AM-1 log M).

(10.7)

0

The integral in (10.7) is 1

1

f f'(x)(x-4) dx = -f(0)+if(1)- f f(x) dx. 0

(10.8)

0

Combining (10.6), (10.7), and (10.8), we have 1

1 f f(x)e(mx) dx = if(0)+zf(1)+O(AM-1logM),

(10.9)

0

and letting M tend to infinity we have the case k = 0, 1 = 1 of (10.1).

A general method of proving functional equations is to write the required function as an infinite series, apply Poisson's summation formula to a partial sum, and then let the length of the sum tend to infinity. This entails a change in the order of summation in a double infinite series. We could prove the functional equations for c(s) and

THE PRIME-NUMBER THEOREM

42

2.10

L(s, X) directly by this method, but it is more troublesome to justify

the interchange of summations and more difficult to identify the functions that arise. We shall therefore prove the identity exp{-(m+8)2iTx-1} = x

00

exp(-m2irx)e(in6),

M= - CO

(10.10)

00

where x is real and 0 < 8 < 1. Rather than apply (10.2) directly, we use eqn (10.9), so that M 1 I f exp{-(r+t+8)21rx-1} dt -M o

= 2 exp{-(r+8)2}+Jexp{-(r+1+8)2}+ + O[rx-1exp{and the double series

I CO

9n=-CO

If

(r+8)27rx-1}M-1 log 112],

(10.11)

r+1

00

exp{-(t+8)21rx-1}e(mt) dt

(10.12)

converges and is equal to W

I exp{-(r+8)2irx-1}. r= - 00

(10.13)

A typical term in (10.12) is 00

f exp{-(t+8)21Tx-1+%rimt} dt - 00 00

=

f exp{-irxz2+2irim(xz-8)}x dz, -00

= xexp{-7rxm2-2Trirn8} f00exp{-1rx(z-im)2} dz,

(10.14)

- 00

where we have put t+8 = xz. The integral in (10.14) can naturally be regarded as the contour integral of a-z' along the line Im z = m. To evaluate the integral, we can move the line of integration to the real axis, when (10.14) becomes 00

xexp(-irxm2-2Trim8) f exp(-lrxt2) dt - Go

= cir-'x}exp(-irxm2-2,rirn8),

(10.15)

CO

where

c = f exp(-t2) dt. - 00

(10.16)

A MODULAR RELATION

2.10

43

If we combine (10.15) and (10.12) we have proved that 00

OD

I exp{-(r+8)27rx-1} M= = -ICO 1- - 00

cvr-4x1exp(-7rm2x)e(m8).

(10.17)

Putting 8 = 0, x = 1 verifies that C = 1TI,

(10.18)

and we have proved (10.10). Equation (10.17) should not be let pass without some comment. Let

us put

00

B(w) _ Y

exp(lrim2w)

(10.19)

na= - CO

for values of co for which the series in (10.19) converges, that is, for complex w with positive real part. Then we have 0(w+2) = 0(w),

(10.20)

and (10.10) with 8 = 0 gives us 02(-1/w) = w02(w),

(10.21)

provided w is pure imaginary. However, since eqn (10.21) holds along the imaginary axis, the two sides of (10.21) have the same derivatives at points iy with y > 0, and, since power-series expansions of regular functions are unique, (10.21) must hold whenever O(w) and 0(-1/w) are both defined, which is whenever the imaginary part of w is positive. From eqns (10.20) and (10.21) we see that 04(w) dw

(10.22)

is invariant under the group of transformations of the upper half of the complex plane generated by w --* w+2 and w -lbw. We are now in the realm of the elliptic modular functions. A modular

function is one that is invariant under the group of transformations generated by co --* w +1 and co - -1 /w or under a subgroup of finite index in this group. The derivatives of a modular function are not invariant under these transformations, since dw itself is not invariant; functions that satisfy the transformation law for a power of a derivative

of a modular function are called modular forms. The name `elliptic modular functions' arises as follows. The periods of an elliptic function

form a free Abelian group on two generators oil and w2. A modular form corresponds to a function of two complex variables wl and w2 which is homogeneous and whose value does not change when we replace wl and w2 by another pair of generators of the same free Abelian group (or, more generally, which takes a finite set of different values when wl

44

THE PRIME-NUMBER THEOREM

2.10

and w2 are generators of the same Abelian group). Here w=W2/wl. Thus we may consider with

f (&J11 w2) = wl'02((02/wl), f (w2, -w1) = w2 1B2(-w1/(02) = wl 102(w2/w1) = f (w1, w2)

and

f (w1, w2+2w1) = wi 162(w2/w1+2) = f (w1, w2)'

(10.23)

(10.24) (10.25)

11

THE FUNCTIONAL EQUATIONS When he awoke in the morning, the first thing he saw was Tigger, sitting in front of the glass and looking at himself. `Hallo I' said Pooh. `Hallo I' said Tigger. `I've found somebody just like me. I thought I was the only one of them.'

11. 21

WE return to c(s) and L(s, X). The definition Go

f(1s) = f

e-vy18-1 dy,

(11.1)

0

valid for u > 0 (we recall s = a+it, a and t real), becomes - s) = 7r18nts f e-mazrxxls-1 dx

(11.2)

0

when we write y = orm2x. Summation over m gives 00

co

m=1

f e-manxx18-1 dx.

(11.3)

Since the sum and integral in eqn (11.3) each converge absolutely, we can rearrange the right-hand side of (11.3) as 00

f

00

e-xls-1 dx =

00

f J(9(ix)-1)x18-1 dx,

(11.4)

where O(w) is the function of eqn (10.19). Next we write 1

co

f J(6(ix)-1)x1a-1 dx

= f (0(i/t)-1)t-1s-1 dt

(11.5)

1

0

and use eqn (10.10) to put (11.5) into the form f J(t10(it)-1)t-18-1 dt

= f 0(it)t- 8-i dt -1(28-1) 1

1 Go

= f (B(it)-1)t-18-1 dt-s-1-(1-s)-1. 1

(11.6)

THE PRIME-NUMBER THEOREM

46

2.11

The left-hand side of (11.3) has now been expressed as M

f

(11.7)

j(9(ix)-1)(x1s-1+x-1s-Y) dx

1

The expression (11.7) was obtained under the assumption u > 1, but, since

6(ix) -1 < e-"x (11.8) for x > 1, the integral in (11.7) converges for all complex s. Since r(s) is a known function, and (r(Zs))-1 is integral (single-valued and regular over the whole s-plane), we can take (11.3) with (11.7) as the definition of i(s), knowing that

m-s agrees with our new definition

when the series converges. We have now continued c(s) over the whole plane. Further, (11.7) is unchanged when we replace s by 1-s, so that (11.9) W-}8r(- sMs) = or18-1r(I-13)x(1-s), the promised functional equation. Since

rM IS)

l1S 2

= 21-sor-4r(s)cos . sor,

an alternative form of (11.9) is (1-s) = 21-8r-sr(s)cos -sor c(s).

(11.10)

(11.11)

We now list some properties of r(s) (see for example Jeffreys and Jeffreys 1962, Chapter 15). The product

r(s+ 1)

= eys

nl a-sins'

co

11,+

,JJ>.-L=111 t

(11.12)

JJ

where y is the constant of (2.5), converges for all s, and defines r(s) as

a function that is never zero and has simple poles at 0, -1, -2,.... Using this information in (11.7) we see that the pole of (11.7) at 1 comes

from c(s), the pole at 0 from r(2s), and that c(s) must have zeros at s = -2, -4,..., to cancel the other poles of r(zs). From eqn (11.12), r(s+1) = sr(s), (11.13) and

r(1+s)r(1-s) = orscosec ors,

(11.14)

where we have used the product formula for sin ors. We can verify eqn (11.10) by showing that the ratio of the two sides is a constant. Equation (11.1) is obtained by evaluation of the limit of w

f 0-1+1(1-t/N)Ndt 0

(11.15)

THE FUNCTIONAL EQUATIONS

2.11

47

in two ways as N tends to infinity. We can also obtain from (11.12) the asymptotic formulae

loge(s) = (s- ))logs-s--21og2Tr+O(1/Isl),

(11.16)

I"(8)/T(s) = logs+0(1/1sI)

(11.17)

and which hold as Isl

oo uniformly in any angle -7T+8 < args < it-8

forany8>0. Next we consider an L-function L(s, X) with X a proper character mod q. There are two cases. If X(-1) is 1, we argue as above up to 00

7r-18q}Sf(zs)L(s, X)

= f xjs-1m=1 I

X(mn)e-"a'ax/q

dx

o

Co

_

f xks-1 y(x, X) dx,

(11.18)

0 co

where

p(x, X) _-coI

(11.10)

X(nz)e-""'Tx/q.

We approach q(x, X) through (10.10): co

00

e-(,"+8)'7+/x = xj I e-"0"xe(mn8).

(11.20)

We put 8 = a/q and use eqn (3.8): (11.21)

X(m)T(X) = Y* X(a)eq(am), amodq

so that

I co

X) _ amodq x(a)

m=-o0

e-"L'',xlgeq(am) ao

I e-(n+alq)'aq/x, !. X(a)(glx)1 amodq M=-00

(11.22)

which we may rearrange as 00

(glx)I-co I amodq G*

R(a)e-("tq+a)'ar/xq = (qlx)}

.4 f'=-00

= (q/x)1pp(1/x, X).

R(r)e-r''alxq

(11.23)

This will play the part of the modular relation (10.10). As before, we split up the range of integration in (11.18) and find that 1

00

f xjs-1p(x, X) dx = f t-js-1(T(7C))-1(gt)lcp(t, X) dx. 0

1

(11.24)

THE PRIME-NUMBER THEOREM

48

2.11

The analogue of (11.7) is now seen to be fCo

Co

Tr I8q 8" (js)L(8, X) = j

xIs-1p(x, x) dx

J

1

x- 8-Iy(x,X) dx.

+2T(X) 1J

(11.25)

As before, the right-hand side of (11.25) converges for all 8, so that

L(s, x) has an analytic continuation over the whole plane, with no singularities. Moreover, L(s, x) must have zeros at 0, -2, -4,... to cancel the poles of 1'(js). We proceed to deduce the functional equation. We have T(X) _

mmodq

X(m)eq(m)

rmodq

X(-m)eq(-m) (11.26)

= T(X),

since it was assumed that x(-1) = 1. By eqn (3.14), since x is proper modq,

(11.27)

q'IT(X) = T(X)Igl.

We now see that the right-hand side of (11.25) is T(x)q-1 times the corresponding expression with s replaced by 1-s and x by X, which gives the functional equation T(X)q-!7r-I+I8gl-I8I'(z- s)L(1-s, X)

-i8gI8I'(zs)L(s, X) =

(11.28)

We now consider characters x(m) proper modq with x(-1) = -1. Since we want to consider a sum from -oo to oo, we use mx(m) in place of x(rn). Writing s+1 for s in (11.2), we have ic8+1)gl(8+1)I'( (s -1))L(s, x) = f00 0

00

me--27T.81gxI8-} dx

m=1 Co

= z f p(x,

&Is-I dx,

(11.29)

0 Co

where

p(x, x) = I

mi(m)e-"z$""Iq.

(11.30)

9,L= - 00

We find a functional equation for p(x, x) by differentiating (10.10) with

respect to S. We get T(X)P(x, x) = ig1x-4P(l/x, X)

(11.31)

Arguing as before, we find -i8-igi8+ P(j(s-F-1))L(s, x) 00

Co

fp(x,

xI8 i dx

1 lqi f p(x, ' & -is dx.

11.32

THE FUNCTIONAL EQUATIONS

2.11

49

Again, the integrals on the right of eqn (11.32) converge for all s, so that

L(s, X) has an analytic continuation; it must have zeros at -1, -3, -5,... to cancel the poles of r(j-(s+1)), and satisfies the functional equation X)

jql

T(X)'-(s+i)g3(S+i)j'((s-F-1))L(s, X).

(11.33)

To check this, we note that when X(-1) = -1 (11.34) T(X) = -T(X). There is also an analytic continuation of L(s, X) when X mod q is not proper. If Xi proper mod f induces X mod q, then for a > 1

L(s, X)

= M=1

X,(r)

µ(d)d 1(d)

Xnm)

4

dim

(d,

fl4 i

(d,f)= 1

(11.35)

f'=1

when we write in = dr. The sum over r in (11.35) is L(s, Xi), which has an analytic continuation since Xi mod f is proper, and the sum over d is defined for all f. The corresponding functional equation for L(s, X) contains the sum over d explicitly. We shall not need this case again.

A number of proofs of the functional equation can be found in Chapter 2 of Titchmarsh (1951).

12

HADAMARD'S PRODUCT FORMULA Suddenly Christopher Robin began to tell Pooh about some of the things : People called Kings and Queens and something called Factors. H. 174

IN proving the prime-number theorem, Hadamard studied integral functions of finite order, that is, functions f (s) regular over the whole plane, with (12.1) log) f (s) I < Is I-d for some constant A, as Is I oo. The order of f (s) is the lower bound of those A for which an inequality of the form (12.1) holds. Hadamard

showed that an integral function of finite order can be written as an infinite product containing a factor s-p corresponding to each zero p of the function. This generalizes the theorem that a polynomial can be written as a product of linear factors. Weierstrass's definition (11.12) of

the gamma function is an example. The product is especially simple when f(s) has order at most unity. The order of 1/1'(s+1) is unity, from (11. 16). We shall obtain the product formulae for e(s) and e(s, X) given by (12.2) e(s) = s(1-s)7r-181'(Js)g(s)

and

e(8, X) = (g1ir)h(3}a)F(z(s-I-a))L(s, X),

(12.3)

where X is proper mod q and a = 0 or 1 according to the relation X(-1) _ (-1)a. Note that (11.9) is just the assertion that E(1-s) is equal to i(s), and (11.28) or (11.33) implies that

If('-s,X)I = l e(84)1-

(12.4)

First we show that ie(s, X) has order one. By eqn (5.3), if a > 0, OD

L(s, X) =

8x-S-1 1

rn r

X(m) dx.

(12.5)

By Polya's theorem (4.2), the sum over mn is bounded, and thus U,

IL(s,X)I
(12.6)

HADAMARD'S PRODUCT FORMULA

2.12

51

and for a > j we have logIL(s,X)I ., and we have

(12.7)

logIe(s,X)I
for a < J. To bound e(s) we recall eqn (5.14): (1-21-s)C(s) _

(12.9)

m=l

valid for a > 0. As in the derivation of (12.7) from (12.5), we deduce that the right-hand side of (12.9) is < IsI for a > 2. Now when a > j we have

(1-s)/(1-21-8) < Isl, logl(1-sMs)I <1ogls1+1, and so and Stirling's formula gives

(12.10)

(12.11)

logle(s)I < 1sl(logls1+1) (12.12) uniformly in a > z. Since e(1-s) = e(s), (12.12) is also valid for a < 2. We have now shown that e(s) and e(s, X) have order at most one. If s is real and positive then (12.7) and (12.12) are best possible, by Stirling's formula, so that the order of e(s) and e(s, X) is exactly unity. We now discuss the zeros p of the functions e(s) and e(s, X). First we prove a lemma.

LEMMA. (Jensen's formula.) Let f (s) be a function of the complex variable s = r cis 0 which is regular in 18 1 < R with no zeros on 18 1 = R,

for which f (0) = 1. Then R

tar

(21T)

f log I If (R cis 0) 1

dO = f r-1N(r) dr,

0

(12.13)

0

where N(r) is the number of zeros of f(s) inside the circle C(r): IsI = r.

f

Proof. We can write the left-hand side of (12.13) as 2ir R (21T)-1

tar R

fRe(df/f)drdo=(21r)_1f 0 r=0

cis 0 drd0 f Re Es) s

0 r=00

R

_ Re f

to

and the inner integral is 2IriN(r).

2ir

1

ff'(s) 0=0

f (s) ds

dr 9'

(12.14)

THE PRIME-NUMBER THEOREM

52

2.12

Now E(O) = 1, so we can apply (12.13) to e(s) at once. We shall prove later that 6(0, x) is non-zero, so that (12.13) can be applied to e(s, x)/E(0, x). To avoid a circular argument, we choose a 8 for which e(8, x) is non-zero

and apply Jensen's formula to E(s, have

x). By (12.7) or (12.12) we

R

f r-'N(r) dr << Rlog R,

(12.15)

0

N(T) < T log T. (12.16) Here, N(T) is the number of zeros of c(s) or of e(s, x) with Is! < T. and as T -> oo

When we examine the formulae (12.2) and (12.3) for i(s) and e(s, x), we see that any zero of e(s, x) must be a zero of L(s, x), and similarly for c(s). The converse is not true, because L(s, x) has extra zeros at negative integer values to cancel the poles of the gamma function in

(12.3). If s = a+it with a > 1, Euler's product formula L(s, X) = II {l-x(p)p-$}-1

p

(12.17)

converges absolutely and so is non-zero. By the functional equation, e(s, x) is therefore non-zero for a < 0, since e(s, X) is non-zero for a > 1. Thus all zeros p = (3+iy of e(s, x) have 0 < / < 1, and the same is true for e(s) by a similar argument. Riemann's hypothesis is that /3 is always 1. Riemann stated the hypothesis for i(s), but it is difficult to conceive a

proof of the hypothesis for e(s) that would not generalize to e(s, x). We shall prove later that 0 < /3 < 1: this statement is equivalent to the prime-number theorem in the form (5.17) in the sense that each can be derived from the other. For later use we now prove a result more precise than (12.16). LEMMA. The number of zeros p = 13+iy of e(s, x) in the rectangle B,

0
T
is at most
(12.18)

(12.19) (12.20)

Proof. Let so be the point 2+i(T+2). Then

IL(so,x)I > 1--a--... > 1.

(12.21)

We apply Jensen's formula (12.13) with R = 3 and (12.22) f (s) _ E(s-s0, x)Ie(so, x). By (12.7) and Stirling's formula (11.16) the left-hand side of eqn (12.13) is

< logq--log(IT I +e).

(12.23)

HADAMARD'S PRODUCT FORMULA

2.12

63

The circle radius 3 and centre s0 clears the box B by a distance at least 1,

so that, if N is the number of zeros required, the right-hand side of eqn (12.13) is > Nlog6/5. (12.24) This proves (12.19), and (12.20) follows similarly.

We shall not need the following more accurate formula. If T > e, the number of zeros of e(s, X) with 0 < # < 1 and 0 < y < T is (2i)-1T log(gT/2lTe)-j- 0(log qT),

(12.25)

from which (12.19) and (12.16) follow readily.

We can now prove the product formula. Let f (s) be e(s, X) or e(s), and let p run through all zeros off (s). By (12.19) or (12.20) the series IPI-2 converges, and so therefore does the product (12.26) P(s) = II (1-s/p)exp(s/p), p#0

where if 0 is a zero we add a factor s at the beginning. P(s) is a regular function with the same zeros as f (s). We should like f (s)/P(s) to be a constant or some other simple function. Certainly (12.27) g(s) = log(f(s)/P(s)) can be defined to be single-valued and regular over the whole s-plane. We shall prove that g(s) = A+Bs. (12.28)

By (12.19) or (12.20), there is a sequence of R tending to infinity with (12.29)

R-IPI > (log R)-1

for each zero p. We want a lower bound for log P(s) on the circle Is I = R.

Now

-

O
logl(1-s/p)exp(s/p)I < B

O
IPI-1 < R1og2R, (12.30)

by (12.19) or (12.20), and #RS ;1p< 2R

log I (1-s/p)exp(s/p) I < 1RS Ip <2R(1--loglog B)

. B log B loglog B,

(12.31)

by (12.29). Finally

-IplI logl(1-s(p)exp(s/p))I
(12.32)

On such a circle we have log If (s)/P(s) I < B log2R.

(12.33)

The left-hand side of (12.33) is the real part of g(s). If we write 8 = R cis 9,

THE PRIME-NUMBER THEOREM

54

2.12

the power-series expansion of g(s) makes g(R cis 0) a Fourier series in 0: g(R cis 0) _ (a(n) +ib(n) )Rn cis n9, (12.34)

Reg(Rcis9) = Rn(a(n)cosn9-b(n)sinn0)

so that

(12.35)

tar

and

jira(n)Rn = f cos(-nO)Reg(Rcis6) I dB.

(12.36)

0

Hence tar

fa(n)IRn << f IReg(Rcis6) dO 0 tar

<< f (IRe g(B cis 0) 1 +Re g(B cis 0) -a(0)) dO 0

<1+

2ir

f max(Re g(B cis 9), 0} d9 0

(12.37) < 1--Rlog2B. Since (12.37) holds for an infinite sequence of R, a(2), a(3),... must be

zero, and similarly so must b(2), b(3),..., and we have proved eqn (12.28). We have therefore e(s) = CBS fJ (1-s/p)exp(s/p), (12.38) P

e(s, X) = C(X)eB(x)s IT (1-s/p)exp(s/P),

(12.39)

P

with the modification mentioned above if p = 0 occurs as a zero. These products converge for all s. By taking logs and differentiating, we get

'(s) C(s)

- B- S-1 -}- z log 27T-1 1'(Is+1)+ P(zs-+-1)

1

1

P

-p -}-1). p

(12.40)

Vp +1), P

(12.41)

and

L'(s' X) L(s, X)

= B(X)-.log 7rq-12I"(2(s+a))+ P(j(s+a))

POO

where p runs over all zeros of c(s) or L(s, X) that do not coincide with the gamma-function poles. We can substitute the equation P(u)

= iY+

(12.42)

which follows from the corresponding product formula (11.13) for P(s), and obtain a sum over all zeros (except p = 0) of c(s) or L(8, X). It can

be shown that

B = log2--4log1r-1-jy,

(12.43)

but no simple expression for C(X) and B(X) is known in general.

13

ZEROS OF e(s) You remember how he discovered the North Pole; well, he

was so proud of this that he asked Christopher Robin if there were any other Poles such as a Bear of Little Brain might discover. 1. 131

WE saw in the last chapter that e($) and e(s, X) have no zeros p = P+iy

with /3 > 1. The prime-number theorem corresponds to the fact that no zeros of ic(s) (these include the zeros of e(s)) have P = 1. All the direct proofs that P < 1 at a zero are based on the following argument. The function has poles of residue 1 at the poles of c(s) and poles of residue -1 at the zeros. Now -S'(3)/4(s) = d

log(1-p-s)-1

P

_

logp(1-P-5)-'.

2)co

_ I A(m)m-, M=1

(13.1)

where A(m) is the function of eqn (1.31). At 8 = 1, the series diverges

to +oo, corresponding to the pole of residue +1. Now, if 1+iy were a zero of (s), we should expect the series in (13.1) to diverge to -oo, and the partial sums to be as large as those for the case s = 1, but negative. To achieve these, the numbers m-'Y must be predominantly near -1. The values of mn2iV are therefore predominantly near + 1, and

there is a pole at 1+2iy with residue +1, and so a simple pole of c(s) at s = 1+2iy, which we know does not occur. To make this argument rigorous, we use (13.1) with s = o+it where o > 1, so that the series converges. For all real 0, 3+4 cos 0 + cos 20 = 2(1+cos0)2 > 0. Since

we have 853518 x

m=1

i;(a)

4 Re

A(ru)m-°cos(itlogm),

i;'(a+it) -Re C'(o+2it) > 0 .

(o+it) E

(a+2it)

(13.2)

(13.3) (13 . 4)

THE PRIME-NUMBER THEOREM

56

2.13

We now make a+it tend to a zero P+iy. Since C(s) has a pole at 1, there is a circle centre 1 and some radius r, within which C(s) is non-zero. (Calculation shows that r = 3 has this property.) If we suppose that

/i > 1-br,

(13.5)

then IyI > br, and so is bounded away from zero. In eqn (12.40),

'(s) = (s)

1

-B+s-1-

log21r-21 P'(18+1)

P(j8+1) -

(+)p 1

1

(13.6)

we shall assume 1 < a < 2, ItI > s . > 0. Here the sum is over zeros p of e(s), not over all zeros of C(s), and, since s-p and p have positive real part, we have Re (s 1

p+pl

>0

(13.7)

whenever a > 1 and p = P+iy has 0 < < 1. By (11.17) the term in P(- s+1) is < log(It I d-e). We now write down three inequalities. Since there is a pole at s = 1 of residue 1, we have

-C'(a)IC(a) = (a-1)-1+0(1).

(13.8)

At s = a-l-iy we omit all terms in (13.6) except those from the particular zero with which we are concerned; by (12.7) this gives us the inequality

I iy) < -(a-P)-1+0(log(IYI+e)). Similarly,

-C'(a+2iy)/C(a+2iy) < O(log(IYI+e)).

(13.9)

(13.10)

When we substitute (13.8), (13.9), and (13.10) into (13.4), we have

4(a-fl)-1-3(a-1)-1 < 0(log(IYI+e)),

(13.11)

valid as a --> 1 from the right. By giving a a suitable value, we see that

R<1

log(IYI+e)'

(13.12)

where c is an absolute constant. If we constrain c to be less than br then, by (13.5), (13.12) also holds when IyI < 4,p- and is thus valid for all zeros p = P+iy of C(s). For twenty years, (13.12) remained the best known upper bound; in 1922 J. E. Littlewood proved that f3 < 1-c2loglog(IyI+e2) log(IYI-I-e)

(13.13)

ZEROS OF C(s)

2.13

57

for some constant c2 (Titchmarsh 1951, theorem 5.17). The latest result is

<

c8(e)

(13.14)

where the constant cs depends one. This was proved in 1958 by Korobov

and by I. M. Vinog adov independently. Intermediate improvements on (13.13) used the intricate methods of I. M. Vinogradov (1954).

14 ZEROS OF e(s, X) `What do you think you'll answer 2V

'I shall have to wait till I catch up with it,' said Winniethe-Pooh.

1.34

IN this chapter and the next we prove results like (13.12) for the zeros p = P+iy of L(s, X), with uniform constants in the upper bounds. We actually work with e(s, X), where X is proper modq; the zeros of e(s, X) are those zeros of L(s, X) that are not at negative integers and so are not cancelled by gamma-function poles. We use the product formula in the differentiated form (12.41), L'(8, X)

- B(X)-11og q-1 ;r

L(s, X)

T'(1(s+a))+,

(

P)w

1 +1

`--' s-P

2 P(21(8+a))

(14.1)

the sum is over zeros p of e(s, X), the term 1/p being omitted if

p = 0. We shall assume throughout the chapter that s = a+it with

1
The first complication is the elimination of B(X) from eqn (14.1). We

subtract from (14.1) its value at s = 2, noting that Co

I L'(2, X) I/IL(2, X) I <

A(rn)mn-2,

(14.2)

na=1

which is bounded independently of X. Estimating the gamma-function term from (11.17), we have

-Re

Re(s 1

L(ss,

P

X))

21

)+0(log(Itj+e)),

(14.3)

where the term p = 0, if it occurs, is now included in the sum. We now

note that

Re 2-

2

logq,

(14.4)

P

by (12.19). Writing we have

2- 2-

1=

P

-Re

1(t) = log{q(Itl+e)}, (s' X)

L(s, X)

<-

Re P

1 + 0(1(t)) s-p

for 1 < a < 2, the implied constant being absolute.

(14.5) (14.6)

ZEROS OF e(o' x)

2.14

59

When we apply (13.2) we obtain the relation Xo) - 4 Re L'(a+it, X) - Re L'(a+ 2it, X2) > 0, (14.7) -3 L'(a, L(a+2it, X2) L(a, Xo) L(a+it, X) valid for a > 1, as the analogue of (13.4). Here, Xo is the trivial character mod q, whose value X(nm) is 1 when (m, q) = 1 and 0 otherwise, and X2 is the character whose value at mn is {X(mn)}2. Although we have supposed X to be proper mod q, X2 might be trivial and certainly need not be proper

mod q. The trivial character Xo is not proper mod q. However, if Xi proper mod f induces X2 mod q, then L'(s, X2)/L(s, X2) and L'(s, Xi)/L(s, Xi)

differ only by terms involving powers of those primes that divide q but

not. f. For a > 1, these terms give at most Xl(m)A(m)

loge

gig

1

1

I logp/(p-1) < log q.

(14.8)

s)lq

The inequality (14.8) applies also for f = 1, X2 = Xo We conclude that (14.6) is valid for any non-trivial X niodq, possibly with a different

0-constant, and that Xo) 1< a-1 - Re - Re L'(s' 18-112 L(8, X0)

1 +0(1(t)) s-p

(14.9)

r for the trivial character Xo mod q. If X2 is non-trivial, substitution of (14.6) and (14.9) into (14.7) gives 4(a-fl)-1 < 3(a-1)-1+0(l(t)),

(14.10)

(14.11) 9 < 1-ci/l(y) implying that for some absolute constant ci when we choose a appropriately. If X2 is trivial, then 4(a-f3)-1 < 3(a-1)-1+(a-1)/{(a-1)2+4y2}-}-0{l(y)}, (14.12) which is consistent with f = 1 when a -> 1. However, if IY1 > c2/l(Y)

(14.13)

for some positive c2, then by choice of a in (14.12) we can show that (14.14) 9 < 1-c3/l(y), with a smaller absolute constant c3. We have now shown that either

(14.14) is true or

lyi < 8/logq,

(14.15)

where 8 is an absolute constant. The absolute constant c3 in (14.14) depends on the choice of 8 in (14.15). When (14.15) is satisfied with

60

y

THE PRIME-NUMBER THEOREM

2.14

0 we can still deduce an upper bound for fi, but it is very close to

unity, and tends to 1 as y - 0. The third and greatest difficulty is to deal with zeros close to unity when X2 is trivial. First we show that there is at most one. We have

-Re L'(a, X) - Re L(a, X)

-

A(m)X(m) ma

00

CO

A(m)m'v (14.16)

If p, - fl1+iy1 and P2 = /32+iY2 are two zeros satisfying (14.15), then

-Re L'(a, X) L(a, X)

-Re

1

(7-Pi

-Re a-P2 1 +O(logq) a-fl2

CF-R

(a-f 1 2+y2

(a-fl2)+Y2 +

0(lo g q )

(a-P )2+32(log q)- 2+ O(log q),

(14.17)

if #2 > a1 > 1-8/(logq). If 8 is small enough, this implies that R1 < 1-c4/l(Y1)

(14.18)

Clearly we can choose e4 < c3, and so (14.18) is true for every zero N1+iy1 of L(8, X) except (possibly) P21 and the possible exception p2 occurs only if X(m) is always real, so that X2 is trivial. Since p2 is also a zero when L(s, X) has real coefficients, if P2 fails to satisfy (14.18) we conclude that P2 is real. We devote the next chapter to the case of an exceptional zero R on the real axis.

15

THE EXCEPTIONAL ZERO Piglet said that the best place would be somewhere where a Heffahunp was, just before he fell into it, only about a foot farther on. 1.57

IN this chapter we consider real characters, that is, characters for which X(m) is always real and thus X2 is trivial. As far as we liow, the corre-

sponding L-functions may have real zeros fi with z < 9 < 1. Just as before we saw that L(s, X) cannot have two zeros both close to 1, we shall now see that two functions L(s, X) corresponding to different proper

characters cannot both have zeros close to 1. Suppose Xl is proper mod q1, X2 is proper mod q2, and the corresponding L-functions vanish at N1 and 92. In place of (13.2) we use (1+X1(mmm))(1+X2(rn)) > 0,

(15.1)

which implies that

X1)-L'(a,X2)L'(a,X1X2) > 0 , X1)

L(a, X2)

(15 . 2)

L(a, Xl X2)

where X1 X2 denotes the character mod q1 q2 whose value at m is X1(?fl)X2(?)Z) When Xl and X2 are different, the character X1 X2 is nontrivial, and (14.6) gives - L'(a, X1 X2)/L(a, X1 X2) < 0(logg1g2),

(15.3)

-L'(a, X1)IL(a, X1) < -(a-(31)-1+O(logg1),

(15.4)

and for L(a, X1)

and similarly for X2. In place of (14.17) we have < 0(logg1g2),

(15.5)

and if Nl > 92 then 92 at any rate satisfies the relation 92 < 1- c1/(log q1 q2),

(15.6)

where cl is an absolute constant. We deduce a uniform zero-free region.

62

THE PRIME-NUMBER THEOREM

2.15

By (15.6) and (14.18) there is a constant c2 with the following property.

Let Q > 1. Then no L-function formed with a character x mod q with

q < Q has a zero p = f+iy with a > 1-c2/log{Q(IYI+e)} (15.7) except possibly at a point Nl on the real axis, where L(s, x) has at worst a simple zero. All x modq with q < Q for which L(f1, x) = 0 are induced by the same real character. To prove the prime-number theorem for an arithmetic progression with common difference q, we need to know that neither c(s) nor any L-function formed with a character x mod q has a zero p = P+iy with p = 1. The proof is simpler if we have /3 explicitly bounded away from -unity. We have to deal only with the case x real, p real. One method is to interpret L(1, x) as the density of ideals in a quadratic number field. This gives a very weak bound. We shall prove Siegel's theorem. THEOREM. For each E > 0 there is a constant c(E) such that if

L(R1, x) = 0,

where x is a character rood Nl < 1 -c(E)q-E.

(15.8)

The constant c(E) in Siegel's theorem is ineffective; that is, the proof does

not enable us to calculate it. All previous constants in upper bounds, such as c2 in (15.7), have been ones we could calculate, given a table of

values of c(s) for Isi < 3 and standard inequalities such as Stirling's formula. Following Estermann's account (1948) of Siegel's theorem, we consider the function (15.9) F(s) = (s)L(s, XI)L(s, X2)L(s, XI X2)1 where Xi and X2 are real characters proper mod q1 and mod q2 respectively. By (15.1), the Dirichlet series d)

log F(s) _

{1+Xl(na)+X2(m)+XiX2(mmm)}A(rn)rn-s

m=1

(15.10)

has non-negative coefficients; it converges for a > 1. For a > 1 we can take the exponential of (15.10) : F(s)

(15.11)

m=1

with non-negative coefficients. F(s) has (at worst) a simple pole at s = 1 of residue (15.12) A = L(1, Xl)L(1, X2)L(1, X1 X2),

THE EXCEPTIONAL ZERO

2.15

63

and no pole if A = 0. Moreover, F(s) has a power-series expansion, co

F(s) = I b(r)(2-s)",

(15.13)

r=0

b(r) _ (-1)"F">(2)/r!

where

_

(-logm)" a (m) -,0.

7'i

(15.14)

na=1

In particular, b(O) = F(2), which is at least unity. The function

F(s)-A/(s-1) _

{b(r)-A}(2-s)"

(15.15)

0

has no singularities, and so its power series converges everywhere. If A = 0, the series on the right of eqn (15.15) is positive on the negative

real axis and represents F(s). Since F(s) is zero at 0, -2, -4,..., we conclude that A 0. We now have f1 < 1. To prove the more precise result (15.9), we use Cauchy's formula, integrating round a circle, centre 2 and radius 2, to obtain the coefficients b(r). On the circle, c(s) is bounded, and for the L-functions we use (12.6), (15.16)

IL(s,X)I <
which was proved from eqn (5.3) and Polya's theorem, and is thus true for any non-trivial X mod q, proper or not. We have 1

lb(r)-AI = 2ai

F(s) ds (s-2)r+1

(s)"(gi log q1) (q21 log q2) (qi q2 log q1 q2)

(15.17)

(.)"g1g21og3g1q2,

the constant implied in the < sign being absolute. We restrict ourselves

to the range i o < o < 1 and estimate the tail of the series (15.13) for F(s). .C.

r=[ci1

00

lb(9)-AI(2-u)1

lg1g2logg3q1q213(2-a))'.

R 00

q1 q2 log3g1 q2

'C`

),.

(16 R+1

glg21og3g1g2(16)R1

(15.18)

THE PRIME-NUMBER THEOREM

64

2.15

and thus (taking only the first term in the series for F(s)) F(a)

a-1

> 1-A

R

1-J

((2__a)R+1-1)/(1-a)-O(gjg2log3q1q2(16)R)

(15.19)

We choose R so that the error term in (15.19) is less than 2; this is possible with a choice of R with

R = Alogglg2, where A is an absolute constant. Now

(15.20)

F(a) > I-A(2-a)R/(1-a) .-A(1-a)-1exp{R log(1+(1-a) )} > 2-A(1-a)-lexp{R(1-a)} j- BA(1-a)-1(q1 q2)'(1-°),

(15.21)

where B is another absolute constant. We now choose X2 so that L(s, X2) has a real zero R2 with

1-e/(A+1) < 92 < 1;

(15.22)

if the choice is impossible then (15.8) certainly holds. Since f (/32) = 0, we have < ql (1-S')) (15.23) a1 < BA(1-N/ 2)-1(q)A(1-fs) l qz where the constant depends on the choice of X2 and so on C. This is a lower bound for the product A of the three L-functions at 1. Since

L(1, X2) is a known constant, we seek an upper bound far L(1, X1 X2) From eqn (5.3), 00 L(s, X) = 1f

8x-9-1 1 X(m) dx. m x

Let r = qi log q, the upper bound in Polya's theorem. Then r

L(s, X) I < f Is I x-°-1x dx + 1

(15.24) (15.25)

(00

fr

Is I x-a-1r dx.

(15.26)

If a > 1- (log q) -1, the first integral is < 18 1 log q and the second is is I, and so

IL(s, X) I < IsI log q.

(15.27)

When we integrate round a circle radius J(log q) -1 to find L'(8, X), we have for a > 1-(lobo q) 1 (15.28)

the bound

I L'(a, X) I < log2q.

(15.29)

THE EXCEPTIONAL ZERO

2.15

65

Hence, if L(s, Xi) has a zero Pi in the range (15.28), A

= L(1, Xi)L(1, X2)L(1, X1 X2) G loggi L(1, Xi)

G logg1(1-R1)L'(a, Xi) (15.30)

for some u in fi < a < 1, and so by (15.23) and (15.29) we have (15.31) 1 G (1-fl1)gi (1-p2)log3gi < (1-fli)gi, where the constants implied depend on X2 and so on e. If Pi does not

satisfy (15.28), then (15.31) certainly holds, and Siegel's theorem (15.8) follows from (15.31).

16

THE PRIME-NUMBER THEOREM The clock slithered gently over the mantelpiece, collecting vases on the way. II. 135

WE can now prove the prime-number theorem in the form (5.17). Combining (5.4) and (5.5), we have 0(ua(Tllogul)-1)

o++iT

f

a-iT

s

s ds

z+0(a/T)

1+0(ua(T logo)-1) On the line a = a, where a > 1, the series

if it < 1, if it = 1, if it > 1.

-0s)Ms) _ j A(m)m-3

(16.1)

(16.2)

m=1

converges absolutely. We now suppose that x is an integer plus one-half. Then a+iT 1

A(m) x8

lie s J a-iT i=1

ds -

!1(m) m <x

0 xa

T

00

A(m) 1

1

l log x/qn l zna

(16.3)

To estimate the series in the error term of (16.3), we first note that

I A(m) ` y,

(16.4)

m
by the sieve upper bound (8.19). Partial summation gives

2 m5ix

+

A(m) ma 11og x/m I

I

m 2.x

A(m) xa log x/zn

`1

a-1

(16.5 )

In the remaining range for m we write in = x--2r, where r is an odd integer. Thus isx <m <2x

A(m) xaloglx/znI

2X-2

<<x-alogx I llogx(1+2r/x)l-1 2x

x-a log x G xl-alog2x.

x/r (16.6)

THE PRIME-NUMBER THEOREM

2.18

67

We can now write the right-hand side of eqn (16.3) as x«

O(x)+C T(a-1)

+xlog2x T

(16.7)

'

where fi(x), as usual, is the sum function of the coefficients A(m) of

-NS)14s)

i ill

C)

I L

Ia

Fia. 2. The contour C

We must now find an approximate value for the integral on the left of eqn (16.3). By (12.20), there is a value of T in each interval

t < T (log T)-1

for which

(16.8)

for each zero p = P+iy of e(s). With such a value of T we move the integral to the contour C consisting of

Cl: the line segment [a-iT, - 2 -iT], C2: the line segment [- z -iT, - 2 +iT],

C3: the line segment [- -+iT, a+iT]. In eqn (12.40),

0s)= B (S)

1

1

1P'(ZS{ 1)

2 r(is+1) +

1

1

s-p+p)'

(16.9)

we subtract the value at 2+it from that at a+it, using the fact that

THE PRIME-NUMBER THEOREM

68

2.16

is bounded on or = 2, where its Dirichlet series converges uniformly. (s)

(s)

= 0(1)-1 r'(+(s+l))+1 r(I+jit)+ 2 r((s+1)) 2 r(2++it)

_

(2+_p) 1

.

1

2-a

= 0(log(Itl+e))+

(or+it-p)(2+i-tp) P

(16.10)
(11.17) on the gamma-function terms. We can now estimate the integral round C. First Ns) xs _

1

2zri

f (s) sof

ds

a

log2T

T

5 xQda -1

<xalog2T/(Tlogx),

(16.11)

and similarly for the integral along C3, whilst 1

f

'(s) xs

(s) s

27ri

d8

x-1log2T f ds s

c2

02

`X-i log3T.

(16.12)

At this stage it is convenient to make the choice of a

a = 1+(logx)-1,

whereupon we have 1 -i7-Ti

f

'(s) xs

ds

(s) s

xlog2T

Tlogx

(16.13)

+x-ilog3T.

(16.14)

The contour has been moved past several poles of the integrand. The are +1 at zeros of c(s), -1 at poles, and so the residues of integral in (16.3) differs from that in (16.14) by Y'(0) Jyj
0

(16.15)

where the sum is over zeros p = fl-I-iy of c(s) with > 0. These are all zeros of i(s), and by inequality (13.12) each satisfies the relation 9 < 1-c1/log(IYI+e).

(16.16)

2.16

THE PRIME-NUMBER THEOREM

69

By (12.20), the sum over zeros in (16.15) is in modulus xi-S

r<21ogT

2 1 T lyl<21

<< xi-8log2T,

where 8 = cl/log(T-]-e). We have now shown that fi(x ) = x }

(16.17)

(16.18)

(16.19)

and when we choose log T

= ( log x) } -J- 0 ( 1 ),

( 16 . 20 )

where the 0(1) is to allow us to find a T for which (16.8) holds, the error terms in (16.19) are bounded by expressions of the form (16.21) < xexp{-c(log x)i} with different values of c. Hence there exists an absolute constant c2

such that

&) = x+0{xexp(-c2(logx)0).

(16.22)

We can quickly deduce from (16.22) that D<x

logp = x+0{xexp(-c3(logx)I)},

(16.23)

and integration by parts gives x

17(x) _ 1 1 = f uu+0{xexp(-c4(logx))},

(16.24)

P<X

2

the classical form of the prime-number theorem. Clearly, only the 0-constants in (16.22), (16.23), and (16.24) are affected when we drop the restriction that x be an integer plus one-half. We have approached the prime-number theorem by the classical

route of Riemann's functional equation and Hadamard's product formula. Neither of these is necessary for the proof of eqn (16.22). There are elementary proofs of eqn (5.17), ones which do not use the complex

variable at all (see for example Hardy and Wright 1960, Chapter 22). The elementary proofs are disappointing in that much extra effort is needed to prove the prime-number theorem with an explicit error term, rather than as an asymptotic equality. An analytic proof that does not use the functional equation is given by Titchniarsh (1951, Chapter 3).

17

THE PRIME-NUMBER THEOREM FOR AN ARITHMETIC PROGRESSION LET

O(x, x) = I A(m)x(m).

(17.1)

m <x

When x is proper mod q, we apply the method of the last chapter to obtain !+ 1 X) = 0 xlog2x _ V(s, X) ds (17.2) I(X ' p 2i L(s, X) s '

f

T) I IV

l>h.

C

where a is given by (16.13), and T and R are chosen so that IIYI-'TI > (logqT)-1,

(17.3)

IIPI - RI > (log q)-1

(17.4)

for each zero p = P+iy of e(s, x). By (12.19), we can suppose that 6 < R < 1. The contour C consists of C,: the line segment [a-iT, - s -iT],

C2: the line segment [-j-iT, -2+iT], C3: the line segment [--+iT, a+iT], C4: the circle centre 0, radius R described negatively. The circle C4 avoids the pole of xG/s and a possible L-function zero at

s = 0. As in (16.10), the inequality I L'(s, X)/L(8, X) I G log2qT

(17.5)

holds on the contour C, the constant being absolute. Thus 1

L'(s, x) x8

Wi

L(s, x) s

c

ds

`Xlog2gT+Xilog2gT. T log X

(17.6)

We now treat the sum in (17.2). By (14.18), non-exceptional zeros satisfy the relation R < 1-cl/loggT (17.7) with an absolute constant c1. If q satisfies an inequality log g < c2(log x)1,

(17.8)

2.17

ARITHMETIC PROGRESSION

71

with c2 < 1, we can choose T so that (17.3) holds, and

logqT = (logx)i-j-0(1).

(17.9)

As in the last chapter, we deduce

- xsl/p1+0{xexp(-c3(logx)°)}

if there is an exceptional

0{x exp (-c3(log x)i )}

if not.

+'(x, X) =

zero /31i

(17.10)

The value of c3 depends on that of c2 in (17.8), but can be effectively calculated.

iT

FIG. 3. The contour U

We can absorb the term xsl/fit only if q satisfies q < loglvx

for some N > 0. By Siegel's theorem with e = 2N-1, N1 > 1-c4q-E > 1-c4(logx)-'&, and xNP1 <xexp{-c4(logx)}.

(17.11)

(17.12) (17.13)

We have now shown that the inequality I'(x, X) I

.

x exp{-c5(log x) l},

(17.14)

where c5 is an absolute constant, holds if (17.11) does, or if (17.8) holds and there is no exceptional zero. 863618 x

THE PRIME-NUMBER THEOREM

72

2.17

Our characters can at last abandon propriety. If X mod q is induced by Xi proper mod f then 0(x, XI)-0(x, X) _

loge

Xi(p'),

(17.15)

an expression whose modulus does not exceed r1a

loge 1+1ogx
(17.16)

The estimate (17.16) also holds if X is trivial modq, b(x, Xi) being fi(x).

This error term absorbs easily into that of (17.14), which thus holds whenever X is non-trivial mod q. For the trivial character Xo mod q, (17.16) gives

I0(x, X0)-X1 <<<xexp(-co(logx)i)+logxp1 1,

and again the first error term will absorb the second. Let ,i(x; q, a) _ A(r).

I

m5x

(17.17)

(17.18)

n=a(modq)

There are two cases. If (a, q) is not unity, only the powers of primes that divide q can occur in the sum (17.18). If, however, (a, q) = 1, we have from eqn (3.3)

/(x;q,a) = ) xmodq

(M)

b(x,X)

_ l?(q)

+0{xexp(-c7(logx)')}, (17.19)

the prime-number theorem for the arithmetic progression with first term a and common difference q. We have proved (17.19) if (17.8) holds and no character mod q has an exceptional zero in its L-function, or if

the stronger- condition (17.11) on q holds. The result (17.19) subject only to (17.11) is called the Siegel-Walfisch theorem. When (a, q) = 1 the arithmetic progression a, a+q,... contains infinitely many primes, but we have not proved that the first x/q terms include a prime unless (17.11) holds, so that x is very much larger than q. For a given q we may have to choose x larger still to ensure that the second term in (17.19) is smaller than the first, and b(x; q, a) is non-zero. Thus (17.19) is useless for numerical work, since we do not know the value of c7. From the proof of Siegel's theorem it follows that, if we knew one L-function with an exceptional zero, we should be able to find the c7 corresponding

to one particular value of N in (17.11). If no L-function has an exceptional zero, then (17.19) holds subject only to (17.8), and we can find c7.

PART III

The Necessary Tools

18

A SURVEY OF SIEVES It wasn't what Christopher Robin expected, and the more

he looked at it, the more he thought what a Brave and Clever Bear Pooh was. I. 140

S u r P o s E that we have a classification of the integers into certain classes, and a system of functions a from the classes to the complex numbers with the properties (i) and (ii) below, which generalize those that the functions (18.1) a(m) = eq(arn) possess for the classification of integers in into residue classes mod q. (i) Apart from the trivial function a for which a(m) is always 1, the suin (or integral as appropriate) of a(m) over a complete set of classes is zero. (ii) If in and n are in different classes there is at least one function a

for which a(m) 0 a(n). Then a sequence mi of integers is uniformly distributed among the classes if and only if

a(mi) = 0(

1(:r(rni) I)

(18.2)

for each a except the trivial one. This is known as Weyl's criterion for uniform distribution. As an example, the prime numbers forma sequence that is -uniformly distributed among those residue classes a mod q for which (a, q) = 1, in the sense that a proportion (1/p(q)+0 (1)) of the primes up to some bound x falls into each reduced residue class. We

THE NECESSARY TOOLS

74

3.18

deduced this from the fact that O(x, X) is of smaller order than x when X is a non-trivial character mod q. Franel's theorem of Chapter 9 fits this scheme. We were concerned there with the distribution of the F Farey fractions in the interval [0, 1]. The appropriate functions a(m) are e(bfm), where b is an integer, and a calculation with Ramanujan sums shows that

r

I

(18.3)

I d111(Q/d),

9n=1

ddb

in the notation of Chapter 9. Franel's theorem relates a measure of the

uniform distribution of Farey fractions in the interval to the mean square of the sums in eqn (18.3) for varying b.

Many results in number theory rest on the uniform distribution of some sequence among certain classes. This is the final step in the proof

of the prime-number theorem for arithmetic progressions. Often the uniformity result is needed at an intermediate stage, and in these cases it is usually sufficient to know (in a quantitative sense) that (18.2) holds most of the time. An assertion that, for a given sequence m1, m2,... and set of classes, eqn (18.2) holds for almost all functions a is called a largesieve result. A common form of large-sieve result is that the mean square of the left-hand side of (18.2) is less than the mean square of the right. The sieve (8.10) of Chapter 8 is such a result with a(m) given by (18.1). In this chapter we shall also consider the characters X(m) and the powers r-8 as functions a(m). These correspond to distribution among reduced residue classes, and distribution of the sequence within intervals. The proof of a large-sieve result depends on an upper bound for an average of some convenient auxiliary function. The simplest example is a result for all characters X to a fixed modulus q. Let u(m) be any complex coefficients and

MI

(18.4)

SX = m=D4+1 u(m)X(M).

In Chapter 7, we introduced the sum S(a) _

DI+N

(18.5)

u(m)e(mo ); ?,z=.31+1

and we obtain the result by comparing the identities

I

amodq

1,3( q

2

2

=q

U(m) bmodq

(18.6)

A SURVEY OF SIEVES

3.18

75

which follows from eqn (3.1), and

u+N ISX12 = p(q) J* ( Xmodq bmodq

(18.7)

u(mn)I2,

nm=D7+1 nt=b(modq)

which follows from eqn (3.3). The sum on the left of (18.6) can be bounded

by the large sieve for exponential sums (7.8), and we have M +N

ISX12 < q-1p(q)(N+0(q-1)) I Iu(nm)12. Xmodq nt=111+1

(18.8)

When we restrict our attention to proper characters we can sum over q on the left of (18.8); by eqn (3.8) for X proper modq, (18.9)

T(X)X(m) _ j* X(a)eq(anz), aniodq

and we have

tll+N

_

(X)SX =

_

.1

Y* X(a)n(m)eq(an1)

nt=31+1 amodq * a mod q

(18.10)

X(a)S\q/.

By (3.3) again we have 2

X(a)S1gl

Srraql

C 9(q)

2

(18.11)

amodq

where the asterisk on the suin over X indicates a restriction to proper characters. When y is proper mod q then IT(X)12 = q, and we have *

L p(q)

q

X

*

ISX12

dq

q

amodq

I s(a)

2

q 11.1 +N

(N+O(Q2))

y

12c(rn)12

(18.12)

nt=e1 +1

by (8.10), that is, by (7.8) again. A third sieve result for characters is possible if the coefficients u(m) are 0 whenever rn has any factor smaller than Q; for then (m, q) = 1, and (18.9) holds for all characters modq. For example, if 112 = 0, and u(m) is 1 when rn is a prime greater than Q and 0 otherwise, we have IT(X)12

q5Q Xmodq

y(q)

ISX 12

N

(N+0(Q2))

Iu(7n)12 1 (N+0(Q2))(1r(N)+O(Q)).

(18.13)

THE NECESSARY TOOLS

76

3.18

The left-hand side of (18.13) can be expanded as

f fs2

Y* Xmodf

µ2(glf)

1SX12

2=iQdf)

(18.14)

y (q)

By eqn (8.16), the term f = 1 gives (ir(N))2log Q(1+0 (1)),

(18.15)

which is approximately half the right-hand side of (18.13) when Q is almost N. If for some X L(s, X) has an exceptional zero in the sense of (15.7), the term in ISXI2 can be as big as (18.15). This give sanother proof that `at most one X has an exceptional zero', but the range for q is not as large as in (15.7) when we work out the details. To obtain a sieve result for a(m) = m-° we return to first principles and use what we shall refer to as Gallagher's first lemma (1967). LEivn1A. Let f (x) be a differentiable function of a real variable x. Suppose that x1,..., xR are at least 6 apart and

X2- 6

X1 { 2b -<-.V,

(18.16)

for r° = 1,..., R. Then

I R

x2

x2

x2

If(xr)I2<Sf If(x)12dx+(f If(x)I2dxl (f If'(x)I2dxl xl

.

ZZ

(18.17)

Proof. We integrate the identity X

I f (d r) I2

= If (x) I2-

(18.18)

If (x) I2 dx

J

T,

over an interval of length 8 and obtain *+48

X,+18

s lf(xr)12 <

Y

If(X) I2 J

Xy

Tr } #$

T,+18

J

J ax If (x) I2 dx dy

J

T,-0

Tr1S

If(x) I2 dx + f

z6 I d If (X) 12

dx.

(18.19)

Tr-s

By (18.16), these intervals are disjoint and subintervals of [X1, X2], and R

x2

x2

Sr I f (xr) I2 < f If(X) 12 dx +IS f 12f (x)f'(x) I dx,

xi

x1

and (18.17) follows from Cauchy's inequality.

(18.20)

A SURVEY OF SIEVES

3.18

77

To obtain large-sieve results from Gallagher's lemma, we arrange the suin on the left of eqn (18.2) to be f (x), different x,, giving the different functions a. For instance, for S(a) given by (18.5) and a(m) by (18.1), the

points xr are the rationale a/q in their lowest terms, and Gallagher's lemma gives lit+N

I

2

S\ci

(Q2+7rN)

q
Iu(mn) I2.

(18.21)

m=111+1

Like the relation (7.8), (18.21) is an approach to the conjecture (7.7). Whereas (7.8) gives N the conjectured coefficient -unity, (18.21) obtains the conjectured coefficient of Q2 but not that of N.

The application of Gallagher's first lemma to m-S was made by N Davenport. Let (18.22) f (t) _ m=1

where a is fixed. The first integral on the right of (18.22) is T2

J

T2

u(rn)

If(t) I2 dt, =,1f T1

T1

!y dt na-it

?n«+it n

in

T2dt I u(m) I2 f rn2a 77L

T2

nitdt. fJ (-1 `9n/

zt(m)i (n)

-F-

99Za92« In

T1

(18.23)

T1

11

111011.

The second term in (18.23) is lit

N

)a

u(nm)ii(n) (n/m)iT2-(n/m)iTi ?nan« i log(n/m)

m#n 2

m=1 pna
+

1 Iu(n)I2

1 Iu(1n)I2

I (n-m)/mI 2

2 ( + n2ni

?112(x + 2 1

2(l

)lo g

n2« ) -I-

Iu(m)m2cxI2

l Iu(n) I2 n2oc ),

+2

2

(18.24)

where we have used the geometric-mean inequality. The coefficient of I u(m) I2m 2a in (18.24) is 0(N log N), and thus we can write the righthand side of (18.23) as

(T2-T1+O(NlogN))

N Iu(rn)I2m-2n.

(18.25)

m=1

Since

f(t) _

N m=1

ilogm

t(m)m-«-it,

(18.26)

and log m < log N, the upper bound for the integral of I f'(t) I2 is at

THE NECESSARY TOOLS

78

3.18

most log2N times that for (18.23). We have now proved that if t1,..., tR satisfy the relations tr+1-tr > 8 (18.27)

for r = 1,..., B-1 and TI

28 < tr G T2- 8

(18.28)

for r = 1,..., B, then R

r=1

If(t r )I2

og < (8-1+1

2

-T

1

f O(Nlog N))

12c(In)I

2

Z-f m o, (18 . 29)

which is Davenport's sieve, first published by Montgomery (1969a).

19

THE HYBRID SIEVE Kanga said to Roo, `Drink up your milk first, clear, and talk afterwards.' So Roo, who was drinking his milk, tried to say that he could do both at once ... and had to ho patted on the hack and dried for quite a long time afterwards. I. 151

IN the last chapter we obtained sieve results in the form of inequalities with two terms on the right-hand side, one of which corresponded to the maximum size of the summands, the other to the mean square of the function times the number of summands. In the series for L(s, X) we can sieve either over X or over s. Montgomery (1969a) sieved over both X and s and obtained a hybrid result, which is better than the one we should obtain by sieving over the one and summing over the other. P. X. Gallagher has found a simple method of obtaining hybrid sieve results, which we describe here. The function f (t) of a real variable t is said to be the Fourier transform of f (x) when

f(t) = I f(x)e(xt) dx.

(19.1)

Under suitable conditions (these include the continuity off at x), OD

f(x)

J'

cit..

(19.2)

-oo

Equations (19.1) and (19.2) imply formally that f I f (t) 2 dt

= f, I (t) f f (x)e(-xt) dxdt f J (x) f f (t)e(-xt) dtdx 00

=

f lf(x)12 dx,

(19.3)

THE NECESSARY TOOLS

80

3.19

Plancherel's identity. We shall use eqns (19.1) and (19.2) with f a linear combination of the values of if IX I < 28,

8-1

F (x )_{0

sin8t, 6t

P(t) =

Here

( 19.4 )

if Ixl>28.

(19.5)

and we can check by means of a contour integral that (19.2) holds except at the points of discontinuity x = ±28. We can verify from first principles that the interchange of integrations in the proof of (19.3) is valid. We now deduce Gallagher's second lemma. S(t) _ ,j c(v)e(vt),

LEMMA. Let

(19.6)

V

where the sum is over a finite set of real numbers v. Let

D(x) =

I

(19.7)

c(v).

IN-xl <(4T)-1

T

ao

f IS(t)I2 dt < 7,2T2 f ID(x)I2 dx. -T -m

Then

(19.8)

Proof. Let 8 = (2T)-1 in eqn (19.5). Then D(x) = 8 1 c(v)F(x-v)

(19.9)

V

b(t) = 8

and

c(v)e(vt)P(t) = 8s(t)F(t).

(19.10)

By (19.3), OD

T

OD

f ID(x) I2 dx = 8 2 5 1 2 ( t ) I2I1' -w -CD and

(t) I2 dt _> 82 f I S(t) I2I F(t) I2 dt,

-T

P(t)I > 2/7r

(19.11) (19.12)

for ItI < T, which gives (19.8). We apply the lemma to a Dirichlet polynomial, that is, a finite Dirichlet N (19.13) S(s, X) _

series

a(m)X(m)m-0-it,

m 1

so that the numbers v are of the form -log m, and Aex

where

D(-x) = IA-18x a(m)X(m),

(19.14)

A = exp((2T)-1).

(19.15)

THE HYBRID SIEVE

3.19

81

The large sieve (18.8) gives 1 a(in)X(m)I <<

((A-A-1)ex+q) j I a(m) I2 a -'ex

xmodq a-1ex

Aex

<(T-1ex+q)dlex I la(m)I2

i (lu( 'r

(19.16)

\')-1

a - (log \')-1

I +(log \')-1

M

FIG. 4. The contour C

By the lemma, T

f

AT

_r Xmodq

IS(s, X) I2 dt < T2 I m-2v 711=1

logniA

f

(T-1ex+q) dx

log(m/A)

N <<

m=1

(m+qT) I a(m) 12M-2o.

(19.17)

A more realistic application is the following. For each character x mod q there is a set of points s(r, x) = a(r, x) +it(r, x) within a rectangle

a
(19.18)

satisfying the condition

tfr+1, x)-tfr, x) > 8,

(19.19)

where 8 is a given real number, not necessarily the 8 of equation (19.4). Clearly we need Gallagher's first lemma (18.17), but it is not directly applicable, since a is also a function of r. We turn to Cauchy's identity. Let C be the rectangle (Fig. 4) whose vertices are

a- (log N)-1+i(logN)-1, 1+(logN)-I±i(logN)-1.

THE NECESSARY TOOLS

82

3.19

Then ifs = a+it satisfies (19.18) we have s2(s '

X)

=

f S2(u+it, X

1

2

n

a+ it -s

)

du ,

(19 . 20)

c IS(s, X) 12 < logN J IS(u+it, X) I2 Idul,

and hence

(19.21)

uniformly in a. We sum over r and apply (18.17) under the integral sign. Next we sum over X mod q. One of the terms on the right of (18.17) is a geometric mean, but Cauchy's inequality allows us to sum over X in each factor before taking the geometric mean. We have now to estimate

two sums over X of integrals, which are of the form (19.17). In each, a and T have been replaced by A and T+18, where u = A+1T, A and r a(mn)m-i7*, and in the second real; in the first a(m) has been replaced by by -ia(m)log in m-iT. It is simpler to describe this inequality in words than to write it out! After applying (19.17) we have N

R(X)

G I S(u+it(r , X), X) I2 << (8-1+log N) nL=1 (mn+gT) I Xmodq r-1

a(m)12m-2A

(19.22)

Finally we integrate round C, noting that c

P

f

m-2a I du

f

l< 7It-2a(log N)-1+

n-

C

+(logN)-'

jjt-2a dA (1ogN)-,

m-2oc min{1-a+ (log N) -1, (log m)-1} (19.23)

< (m2alog(mn+e))-1.

Hence

N

R(X)

tog N

I I S(s(r, X), X )12 < (8-1+log N) I (mn+qT) log(in+e) Xmodq r=1 L=1

I

a (2rn)1 2 ?n

(19.24)

where for each X the points s,. = s(r, X) satisfy (19.18) and (19.19). We can use (18.12) in place of (18.8) and show that T

N

dt IS(s,X)12

f

-T "Q fi(g) Xmodq*

(1n+Q2T)

(19.25 )

9,L=1

and R( ) IS(s(r, X),

q5Q Y(g) Xmodq r=1

X) 12

N

(m+Q2T)

<(8-1+logN)

fl=1

logN la(m)12 log(m+e) MU, '

where the points s(r, X) satisfy (19.18) and (19.19) for each X.

(19.26)

3.19

THE HYBRID SIEVE

83

The point of these hybrid sieve results is that we have N+Q2T (in fact, m+Q2T) in (19.25) etc. in place of (N+Q2)T or (N+T)Q2 which we would have from sieving one parameter and summing the other. A useful mnemonic for large-sieve results is that N Y ja(m) I2nz-2° corre-

sponds to the square of the trivial term t = 0, X trivial, and that there are O(Q2T) non-trivial terms, whose mean square is E Ia(m)j2n,-2o. We have certainly shown that eqn (18.2) holds for almost all pairs s, X.

20

AN APPROXIMATE FUNCTIONAL EQUATION (I) In a corner of the room, the tablecloth began to wriggle. Then it wrapped itself into a ball and rolled across the room. Then it jumped up and clown once or twice, and put out two ears. It rolled across the room again, and unwound itself.

II. 135

W u now have efficient tools for averaging finite Dirichlet series. We would like to average powers of L(s, X) over points s and characters X. In this chapter and the next, we express powers of L(s, X) as finite sums plus error terms. The best results of this type contain partial sums for L(s, X), as one would expect, but also partial sums for L(1-s, X). They have thus earned the nickname of approximate functional equations. There are many ways of proving approximate functional equations; we give a straightforward method of H. L. Montgomery's which uses the functional equation explicitly. Our application leads us to consider L2(e, X), where X is proper mod q. This has the functional equation L2(1-u, X) _ (q/-)2i-1G(u)L2(u, X),

(20.1)

in which we have put G(u) = G(u, X) =

q

T2(x)

I'2 12(u I a) r2(z(1-u+a))

(20.2)

where a is 0 if X(-1) is 1, 1 if X(-1) is -1. We shall use u for a complex

variable it = A+ir, where A and r are real. By Stirling's formula (11. 16), when it tends to infinity in a region A < ITI we have Jr(u)I = (27T)llit la-ie-i IT1(1+O(IT1-1)),

(20.3)

and so under the same condition on u, (zglTI/ir)2,1-1

(20.4)

We use the ideas of Chapter 5 to get a partial sum for the Dirichlet series of L2(s, X). In the fundamental equation (5.4) the convergence of

AN APPROXIMATE FUNCTIONAL EQUATION (I)

3.20

85

the integral is only conditional, so we introduce a kernel K(w) to make the convergence absolute. Let I.1(w) =

327T4ew{(w-27Ti)(w-i7ri)(w+i7ri)(w+ 2

14cosh w rj (w-(r-2)7ri)-1.

K(w) =

and

(20.5)

7ri)}-1

-1

(20.6)

We have normalized so that K(0) is 1. In partial fractions, Ic1(w) w

l e0 2w+

9

2

eW

32(2-r)!(1+r)! w-(7

2+1o

xW K(w)

1

27I

J

;;

w

do

(20.7)

2)ai

m =c-,

(20.8)

x

2-ioo

where

c(v)=0 ifv>e if e-1 < v < e

c(v) = c(v) = 1 if 0 < v < e-1

.

(20.9)

Let s = a+it be a fixed complex number with 4 < a < z . The restriction on a is not essential for the proof, but it is sufficient for our application. Then d(m)X(m)cm =. ..

In<ex

1713

x

1

x1,'K(w)

21Ti

W

L2(s+w, X) do.

(20.10)

By construction K(w) is an integral function, and the only pole of the integrand in (20.10) is at to = 0, with residue L2(s, X). The difference between L2(s, X) and the sum in (20.10) is therefore given by an integral along a contour passing to the left of this pole, which we shall take as

the vertical line Re o = - z -a. Equations (20.1) and (20.4) provide bounds for the integrand that justify moving the contour: for example, they imply that I L(- I +iT, X) I2 < g2T2,

(20.11)

since L(3-iT, X) is bounded, and that IK(A+iT) I < T-4

(20.12)

in -1 < A < 2, I T I > 10, so that the integral along Re w = - -1 - a converges absolutely. We change the variable of integration to u

THE NECESSARY TOOLS

86

3.20

(= A+iT) where s+w = 1-u, and use the functional equation (20.1) to write d(rn)X(m)

X)-

L2(s,

c (7111)

?ns

_

j+Im 1

x1-8-'LK(s +U-1) q 2"-1 G(u)L2(u,

s+u-1

J

)

or

X) du.

(20.13)

Because of the factor K(s+u- 1), the integrand in (20.13) is largest when a is close to 1-8. Now when it is near 1-8 the terms (20.4) approximate the (2A-1)th power of the constant Jq Is I 17r. In modulus at any rate the main contribution to (20.13) is an integral of the form (20.10)

with s and X replaced by 1-s and X, and x replaced by y, where the integer y is given by y = [g2t2/(4,.2x)]. (20.14) More precisely, provided IIm(s+u-1)1 < It 11, we have 1

g

2tt-1

iITII

Cr(26) Cl

yl-u I

It11)2)A-1

(g2(ItI -I41T2xy \

< (1+0(ItI-1)+O(Iyl-1))2jtj¢ < 1,

provided that If we choose y by

(20.15)

y' ItIl.

(20.16)

4zr2xy = q2t2

(20.17)

exactly, so that y is not necessarily an integer, the term 0(y-1) is not present and (20.15) is true in any case; but in this application we shall have y > gItl and (20.16) will hold easily. The series for L2(3+ir, X) converges absolutely uniformly, so that we

may integrate it term by term. By analogy with the term-by-term integration of (20.10), we break the series up into Pi+p3 or into PP2+94

(according to whether we take K1(s+u-1) or K1(1-s-u)), where Y11 .., Y4 are given by

Pl(u) _

d(m)X(m)m-u

wKev

rP2(10 _ m<0-1v

p3(u) _ni>Iey 'P4(u) _

(20.18) d(m)X(m)m-4r

d(m)X(m)m-

m>a-,

We write the integral in (20.13) as the sum of four integrals numbered correspondingly h,..., 14. First we consider the integral I3 involving

AN APPROXIMATE FUNCTIONAL EQUATION (I)

3.20

87

p3(u)IK1(s+u-1). When (20.15) is valid, the integrand decreases in modulus as we move the contour to the right. We employ the contour C consisting (in the case ItI > 10) of

Cl: the line segment (2-ioo, 2-it-iltli], C2: the semicircle centre 2-it, radius It Ii to the right of the line A = 2,

C3: the line segment [2-it+iltli, 2+ioo),

2

- it

I -S

x

Fla. 5. The contour C

which is the line A = 2 with an indentation around the poles of K1(s+u-1). If Its < 10 we take C to be the line A = 2; the estimates below will then hold with ItI-' replaced by 1 and with different implied constants. 2 we have For A

I c m>oy

_ rn> oy rn-{D(m)-D(m-1)}

(20.19) C (ey)1-Alogy, by partial summation and the estimate (2.13) for the sum function D(m) of the divisor function. By (20.19) and (20.4), 1 Y7-Ti

f

J

xl-3-UKj(s+11-1) M 2u-1

8+u-1

7r

G(u)cp3(u) du

Cl

xl-a-A

f

q j r 12A-1 yl-A log y d r

(1--It+.rI)6( 2ir)

ClJ

1-1

G x- t

2

(4q t2t2)

qx-Q It 1-1log Y. 853618 X

G

4I t I logy

(20.20)

88

THE NECESSARY TOOLS

3.20

The same estimate holds for the integral along C3. On the semicircle C2, (20.4) and (20.15) are valid, and we have the upper bound x-0'(1+ ItI')-5gltI log y(iIt 11)

qx-0'lt1-1logy (20.21) Hence I. and similarly 14 are bounded by the expression in (20.21).

21

AN APPROXIMATE FUNCTIONAL EQUATION (II) `Why, what's happened to your tail?' he said in surprise. 'What 7xas happened to it 7' said Eeyore. `It isn't there!' 1.43

WE have now shown that 13 and 14 can be regarded as error terms. We treat Pi and cP2 by moving the contour the other way. The first case

considered is Itl > 10. If t > 10 we use the contour D given by

Dl: the line segment (-I-ioo, -+-it-iltIA], D2: the semicircle, centre -J-it, radius Itll, to the left of A = -,

D3: the line segment [-J-it+iltl1, -i-i], D4: the line segment [-i-i,1-i], D6: the line segment [1-i, 1+i],

D6: the line segment [,'+i, -+i],

-2-ioo).

D7: the line segment

The contour D for t < -10 is the reflection in the real axis of that described above. The indentation about the origin avoids a possible double pole of G(u) at u = 0. The analogue of (20.19) is that, uniformly in A < 1, (ey)1-Alogy. (21.1) Ipi(u) I <<

Let

I1 = 21

f

_ri D

al-s-5+ (8

1 -1)

2u.-1

G(u)p1(u)

du.

(21.2)

(g/J

Using (21.1) in place of (20.19) we see that

IIiI <
(21.3)

and the right-hand side of (21.3) is similarly an upper bound for 12 in which K1(1 -s-u)cp2(u) replaces K1(s+zv-1)cp1(u).

THE NECESSARY TOOLS

90

3.21

Between the contours C and D lie the poles of w-lIC(w) at w = 0 and

to = ±(r-z)7ri, where w = s+u-1. The residues at these can be calculated term by term; in total they give (21.4) mu5ey

1

-it

-.c

9<

Fia. 6. The contour D

where c'(v) = 0

c'(v) -

if v > e

2(x)1-zs

9 -1

G(1-s) I1

32 (1+r)! (2-r)! ,r2vxy)(r_i X

X G(1-s+(r-2)1ri) c'(v) =

q2e

q)1-2s

(21.5)

if a-1 < v < e

G(1-s) if 0 < v < e

If X(-1) is -1 then G(u) is regular at 0. For Its < 10 we take the contour D to be the line A = - . The bound for Il and 12 on this contour

3.21

AN APPROXIMATE FUNCTIONAL EQUATION (II)

91

is the same, but without the factor It1-1 and with a different implied constant. If X(-1) is + 1, however, there is a double pole. If a is bounded away from 1 we can run the contour D between the origin and the poles of K1(s+u-1)/(s+u-1). If a < 1 we take D as D1: the line segment (- z -ioo,

D2: the line segment [-I-i, D3: the line segment [i(1-a)-i,. (1-a)+i], D4: the line segment [1(1-a)+i, --I+i], D5: the line segment [-J+i, -J+ioo). The upper bound for the integral along D3 will be

qx-°(1-a)-2log y.

(21.6)

and get an If all else fails, we can take D as the vertical line A explicit but very complicated extra term in the approximate functional equation from the residue at u = 0 of the multiple pole. When It I < 10, (20.14) is interpreted as

q2 <xy
(21.7)

We have now shown that an equation of the form L2(s, X) =

holds both in

d(971)X(9n 3

e

qn

+

d(gn)X(gn) 1

9n C

/

+I1+I2+I3+I4 (21.8)

< a < 4, ItI > 10, in which case each of Il,..., 14 is

<

q2-0, I t I -flog y,

(21.9)

and in 1; < a < 1, ItI G 10, when each of Ii,..., 14 is (1-a)-2qx-° log y,

(21.10)

the estimates being uniform in the ranges stated. Although we have assumed q > 1, so that L(s, X) is not c(s), we can treat 2(s) by the same method. We assume ItI > 10, since it would be

unprofitable to try to approximate near the double pole at s = 1. When we move the integral in (20.13) to the line Rew = -i-a, this double pole gives a residue

Mw-'K(w)+aw (w-1K(w))

(21.11)

evaluated at to = 1-8, where A is the second coefficient (in fact, A is Euler's constant y) in the Laurent expansion

(s) = (s-1)-1+A+0(s-1)

(21.12)

92

THE NECESSARY TOOLS

3.21

of (s) about the pole at s = 1. We must add to eqn (21.8) the additional terms (21.11); they are ItI-5 (21.13)

<

uniformly in I t I> 10, 1< < 4. There are many approximate functional equations in the literature. The one we have proved is easy to sieve, but has more complicated coefficients. Approximate functional equations for c(s) and for 2(s) are given by Titchmarsh (1951, Chapter 4), and generalized by Chandrasekharan and Narasimhan (1963). A. F. Lavrik has a number of papers on the subject in the Doklady and Izvestia of the USSR Academy of Sciences. Fogels (1969) has a form rather like that of Montgomery's given here.

22

FOURTH POWERS OF L-FUNCTIONS They were out of the snow now, but it was very cold, and to keep themselves warm they sang Poole's song right through six times, Piglet doing the tiddley-pours and Pooh doing the rest of it, and both of them thumping on top of the gate with pieces of stick at the proper places.

II. 7

THE approximate functional equation for L2(s, X) is used to obtain upper bounds for J L(8, X) I at individual values of s, or on average. Our inspiration is the Lindelof hypothesis, (22.1)

I L(J+it, X) I < (qt)E,

for each s > 0, with a constant depending on E. In this chapter, we show that the mean fourth power of L(s, X) satisfies the inequality (22.1). Beyond the fourth power, the sums in the approximate functional

equation are too long for us to prove (22.1) even on average. We use the hybrid sieve (19.26), q

252

(P(q)

R(X)

I S(s(r', X), X) I2

Xino(lq r=1

(8-1+logN) >" (urn+Q2T) ?n=1

In

In(m2) 12

AT

l og(',n+e)

', 2a

.

(22.2)

Although (22.2) allows us to vary a, we take v = z throughout, since (22.3)

1

the gamma functions in G being evaluated at conjugate for a complex points. The proof is no simpler, but the form of the upper bounds is less complicated. Cauchy's inequality applied to eqn (21.8) gives I L(s, X) I4 G 61 1 d (nz)

X(m)m1-8c(m/x) 12+ 4

+61 1 d(m)X(m)ms-1ct (m/y)I2+6r1 II,.I2,

(22.4)

THE NECESSARY TOOLS

94

3.22

and we have a similar result for ic(s), with 7 instead of 6 and an extra term O(It1-10). The integers x and y in (22.4) are connected by (20.14): (22.5)

y = [g2t2/(47r2x)].

When we fix x and average over X and t, y is varying. For a good upper

bound, x and y must be of the same order of magnitude. We restrict

ourselves for the moment to P < q < 2P and U < I t l < 2U, and average the right-hand side of (22.4) over all integer values of x between For each fixed X and t the corresponding values IPU/7T and of y given by (22.5) are distinct and lie between JPU/7r-1 and 16PU/7r. The average of the square of the terms involving y taken over all integers y in this range is at least 4 times the corresponding average over the values of y that actually occur in the sum. This device allows us to sum a Dirichlet series of fixed length ey over varying X and t, provided that x does not occur in the coefficients (and vice versa). For each value of x in the above range, (22.2) gives 2PU/ar.

V*

1

p
CD(q)

X

q

R(X)

2

d(m)X(m) c(m

ZI
fn<ex

m8(r',X) ex

<(8-1+logx)

) d2(m)lo

(m+P2U)mlog(m,-}

P2 U log5(P U+e),

',;

e)

(22.6)

where we have used (2.24) for I d2(m74)/m,, and partial summation. We have assumed 8 > 1. (22.7)

It is only slightly harder to average over the reflected series. For e-1y < m < ey we must break the coefficients c'(m/y) up into five terms corresponding to the five poles of 1K1(w)/w. We take out a factor 01-2TT2x)-r7ri eq2

G(1-s+rai)

(22.8)

from the term corresponding to the pole of K1(w)/w at rori, where

r = 0,

2. By (22.3), the modulus of the expression (22.8) is fixed 2, independently of x, q, t, and the parameter a in the definition of G(u),

which is 1 if X(-1) _ -1 and 0 if X(-1) = +1. After this factor has been removed, a sum like that in (22.6) remains, with no gamma-function

factors and no concealed dependence on x. For each r, the right-hand side of (22.6) is an upper estimate for this sum, possibly with a different 0-constant.

FOURTH POWERS OF L-FUNCTIONS

3.22

95

The contour integrals I1,..., 14 present complications. First we fix t and average only over characters. We have for any function F(u) for which the integrals exist

f F(u) du 2 <

I

\D

,1

DJ

Idol 18+1c-1 2J/ \

f IF (u)I2Is+u_112 Idul/

D

(22.9)

and similarly for integrals along the contour C. The first factor here is I t I-!. We shall find an estimate for the average of I. Here, xl-s-1t(s+u-1)-llK1(s+u-1)(q'/ir)2it-1G(u,

F(u) =

X) PI(21).

(22.10)

We wish to average this over X with the length of the sum P1(u) fixed, that is, with y fixed. By (22.4), x is approximately varying as q2, and (q2/2c)A is bounded on the contour D by a function of t and y alone, by (20.14). Similarly, (20.4) permits us to take out (q/7r)2 11-10(u) at its maximum. By (18.12), q

P
I(pl (u)1 2

< (y+ P2)

d2(2n)

%

mts ev

<

I!L2A

(y+P2)(ey)1-2a

log3y,

(22.11)

where we have used the estimate d2(m) < log3M,

(22.12)

which can be deduced from (2.24). The sum of the integrals involving F(u) on the right-hand side of (22.9) is now tF

max max

(It11)8

(max

/v\2-2u/\4u-2 2 2a I -) 02(u)(y+P2)(ey)1-2Alog3y (--

X)1-2a It 1-1(y+P2)log3y

< (P2+PItI)ItI-11og3(P(ItI+e)), where we have used (20.15). Hence q

P
(22.13)

L II112 <(P2+PItI)ItI-4log3(P(ItI+e)), modq

(22.14)

and similarly the same upper bound holds for the corresponding sum involving 12.

THE NECESSARY TOOLS

96

3.22

To apply a Gallagher sieve we need a similar bound for a sum involving It

1,.. The contour D varies with t, but this was only to help estimation,

and we can keep D fixed and differentiate the integrand. A similar calculation will now give

d1

q

at- 11

P
2

G (P2+PItI)ItI-4log6(P(Itl+e)). (22.15)

Gallagher's lemma (18.17) gives 2U } }

P

f

f 11112 dt+

11112 < ,.=1 U
2U+}

2U+ 11112

U-}

U-1

Ii

dt12(

/

J

U-}

at ,-I,(22.16)

We sum (22.16) over X (using Cauchy's inequality on the second term) to find q

P
1112 < (P+U)PU-3log4(P(U+e))

(22.17)

when we substitute in (22.14) and (22.15). The same upper bound holds for the sum with I2. The integrals 13 and 14 present a further complication in that T. and p4 are not finite sums. To treat p3 we write m>ey

d(m)X(m)m-

12 <

00

n-2) ( `n

1

1

n=1 00

7121

a"y
(22.18)

By (18.11) we have P

q

P
o"U<mS a"+iy

mU

G (en+1y+P2)(eny)1-21(n+logy)3,

(22.19)

and the sum of terms of the form (22.19) from n = 1 to infinity converges

uniformly in l to

< (y + P2) (ey) 1-2A log3y.

(22.20)

When we use (22.20) in place of (22.11) we show that the sums over 13 and 14 also satisfy the bound (22.17).

FOURTH POWERS OF L-FUNCTIONS

3.22

97

We have now shown that R(X)

q

P<2-<2P

eD(a)

Xmodq

L

IL(z+it(r', x), x)14

r=1

U
<
term U-s, which is easily absorbed in the second term on the right of (22.21). When we sum over values of P and U running through powers of two, we can show that q

R(X)

q5Q o(q) Xmodq =1

I L(z+it(r', X), X)I4 < Q2T log5QT,

(22.22)

provided that for r = 1,..., R(X) we have

-T < t(r, X) < T,

(22.23)

and for r = 1,..., R(X)-1 we have

t(r+l, x)-t(r, X) > 1,

(22.24)

and for q = 1 I t(r, X) I > 1. We could replace (22.24) by a weaker condition (22.25) t(r+1, X)-t(r, X) > (log QT)-1, and still have the upper bound (22.22), so that the root mean fourth power of L(s, X) is < log QT. Without the sieving over T this method

proves that the root mean fourth power at a fixed s is <
A simpler proof of a fourth-power moment is in Gallagher (1967). Another result is given by Linnik (1964, section 41). These relate to summing over X. The situation is much easier if we merely integrate over t; there are examples in Titchmarsh (1951, Chapter 7). The hybrid moment we have proved here is sketched in Montgomery's paper (1969b), but with a larger logarithm power.

PART IV

Zeros and Prime Numbers

23

INGHAM'S THEOREM He tried Counting Sheep, which is sometimes a good way of getting to sleep, and, as that was no good, he tried counting Heffalumps. I. 62

WE shall estimate the number of zeros p = 13+iy of L(s,X) or of (a) in a rectangle a < a < 1, --T < y < T (23.1)

where a > 2, T > 1. Let X be a large integer and 111(s, X)

=mIXt

'(M)X(M)m-8.

(23.2)

If Riemann's hypothesis were true, then for a > z 111(x, X) would tend to {L(s, X)}-' as X tended to infinity. We write L(s, X)-B1(s, X) = l +f(8, X),

(23.3)

where f (s, X) has the Dirichlet series a(rn)X(m)9n-1,

a(m) _

µ(d).

(23.4) (23.5)

The series (23.4) converges without any hypothesis for a > 1. Until A. I. Vinogradov's work (1965), the number of zeros was customarily estimated by integration of the logarithm of 1 + f (s, X) round the boundary of the rectangle (23.1). As we shall see below, f (s, X) 'is less than unity in root mean square, whether averaged over t or over X or

INGHAM'S THEOREM

4.23

99

over both, provided a > 1. At a zero, f (p, X) is -1. A. I. Vinogradov revived an alternative approach: to count directly the number of times f (s, X) has modulus at least unity. While no simpler in detail, this method is the more flexible, and we follow it here.

The integral transform 2+i eo

w

dvv = e-m!Y

f 2-ico

(23.6)

is easily verified by moving the line of integration to Re w = z - R, where R is a large positive integer, and letting R tend to infinity: the residues at poles of I'(w) give the terms of the exponential series. Hence

e-l!Y+r>X f

a(9n)X(m)m-Pe-111/I'

2+i o0

_

1

L(p+w, X)M(p+w, X)Ywr(w) dw.

21ri

(23.7)

2-ieo

Here p is a zero of L(s, X) so that the zero of L(p+w, X) cancels the pole

of r(w) at w = 0. We take the integral back to the line Re w If X is not trivial, the right-hand side is equal to i-P+100

f

L(p+w, X)M(p+w, X)Ywr(w) dw,

(23.8)

>}-fl-100

and if X mod q is trivial there is an extra term p(q)q-1M(1,

X)Yl-P11(1-p)

(23.9)

from the pole at p+w = 1. (23.10) l = log QT We write log Y < 101. (23.11) and suppose that log X < 101, The terms on the left of (23.7) with m > 1001Y contribute less than .,

if l is sufficiently large. We recall Stirling's formula in the form (20.3), valid if A < ITIi: tr(A+iT)I = e-i'r1T1IA+iTIA-i{(27r)I+O(ITI-')}.

(23.12)

The term (23.9) is now seen to be at most io when lyi > 1001, again

provided l is sufficiently large. Apart from the zeros of c(s) with lyj < 1001, all zeros fall into one or both of the following classes.

Class (i). Zeros p with I

I

X
a(9n)X(m)9n-Pe-"11Yi

> a.

(23.]3)

ZEROS AND PRIME NUMBERS

100

4.23

Class (ii). Zeros p with i-S+ioo _ iStco

L(p+w, X)11I(p+w, X)Ywf(w) dwl > 11r.

(23.14)

We subdivide class (i) by writing the range (X, 1001Y] as the union of intervals Ir: 2''Y < m < 2''+1Y (the first and last intervals having instead the end points X and 1001Y). A zero is of class (i, r) if

I

a(mn)X(r)mn-Pe-m1YI

> {20(r2--1)}-1.

?fl Ir

(23.15)

By (12.19) or (12.20) the number of zeros of a fixed function L(s, X)

in a subrectangle t < y < t+1, a < P < 1 of (23.1) is 0(l). From each class of zeros in (23.1) we can pick out a sequence of zeros whose imaginary

parts differ by at least unity, in such a way that the sequence contains a proportion of at least > l-1 of the zeros of L(s, X) of that class in (23.1). It is possible that our sequence contains all p at which L(p, X) has a zero of the given class, but these p are multiple zeros of order 0(l). We write 11i(X) for the number of class (i) zeros of L(s, X) in (23.1), N2(X)

for class (ii) zeros, and N(X) for the total number. We call our subsequence the representative zeros. Summing over representative zeros p of class (i, r) we have q

a(rn)X(m) a

*

Xmodq p

q_< Q

`1

9fl I,

mP

?flEI,

l(2''Y)1-2a(Q2T+2''Y)Iog4Y exp(-2'')

< l5

exp(-21')(Q2T(2ry)1-2a+(2rY)2-2(x),

(23.16)

by (19.26) with 8 = 1; we have used (23.3) to estimate a(m) by a divisor function, then (2.24) and partial summation. Comparing (23.15) and (23.16) and adding l2 for the zeros of c(s) with IyI < 1001, we have

(I 5Q

y(q)

N ( X ) < 12+16 N

(r2+1)exp (-2r) X

Xmodq

X

{Q2T(2''Y)1-20C+(2rY)2-2a1 < Y2-20L16,

(23.17)

where we have assumed X > Q2T. We now choose

X = Q2Tl. (23.18) There are several ways of treating zeros of class (ii). The ingenious work of Montgomery (1969b) makes much use of this flexibility. To

INGHAM'S THEOREM

4.23

101

obtain Ingham's theorem we raise the expression on the left of (23.14) to

the four-thirds power and sum over representative zeros p = P+iy of class (ii). By Holder's inequality we have

1±i (P(q)

/

Y2(1-2x)131

I`

*

L(p+w, X)111(p+w, X)yt p(w) dwla

II

Xmodq p

4

1-V ioo --ioo

q 2* f J q
I L(2L+iy, X) I4I P(2L-N) I1 Idu I) 3 X

`2

}+ioo

X(

q
y(q)

f

I M('u+iy, X) I2IP(u-/3) I1 Idul

.

(23.19)

Xmodq P J-ioo

By (23.12) and (22.22) the first integral on the right of (23.19) is 1

C f Q2Tl5max(t+f3-2)-t dt+ P 00

+ f Q2(T+t)(l+log t)5

max(e-27Th13t-4P13) dt

P

1

(o -j)-§Q2T16.

(23.20)

When we apply the hybrid sieve (19.26), the second integral on the right of (23.19) becomes (m+Q2T) logX < log(m+e) m

2

,n=1

*

(a-' z) 4Q2T12,

(23.21)

by (23.18). The right-hand side of (23.19) is (a-+)-1Q2TY2(1-2x)1313,

and thus

.A2(X) < (a- 2)-1Q2TY2(1-2x)1314.

q5Q

(23.22) (23.23)

Xmodq

Y=

We now choose

(Q2Tl-2)31(4-2a))

(23.24)

so that (23.17) and (23.23) together give q

* 11T(X) < (a-2)-1(Q2T)3(1-x)1(2-x)161(2-a)

q
(23.25)

ZEROS AND PRIME NUMBERS

102

4.23

We have assumed that 1001Y is greater than X; this is certainly

true if

a>

1 +41-11og 1.

(23.26)

If a is less than this bound we quote (12.19) or (12.20), which give an upper bound N(X) C Ti, (23.27)

f N(X) < Q2T1

so that q

(23.28)

(P(4) xmodq

in any case. In fact, (23.27) and (23.28) give K N(X) q
z min{Q2T1, (a-

j)-$(Q2T)3(1-(x)1(2-0)161(2-a)}.

(23.29)

Ingham proved his theorem for the zeta function only, with no sum over X; his result (1940) was

N<

T3(1-0)1(2-a)l5,

(23.30)

where N refers to the zeros of the zeta function satisfying (23.1). (23.25)is

stronger than (23.30) for a near 1, but weaker for a > 6. We have lost a logarithm factor by taking representative zeros instead of counting according to multiplicity.

24

BOMBIERI'S THEOREM `I see, I see,' said Pooh, nodding his head. `Talking about large somethings,' he went on dreamily, `I generally have a small something about now-about this time in the morning.' 1.48

WE proved the prime-number theorem for arithmetic progressions a mod q uniformly in q < Q, where Q = (logx)N,

(24.1)

in the notation of Chapter 15. In 1965, Bombieri (1965) and A. I. \Tinogradov (1965) proved independently that the prime-number theorem holds uniformly for almost all progressions with Q a little smaller than xl. Bombieri's result was the following theorem.

THEOREM. Given A > 0 we can find B > 0 such that for all large x

max ' max 0(y; q, a)

Q amodq ii x

-

x

(log x)d'

cPq)

Q = xI(logx)-B

where

(24.2)

(24.3)

and the asterisk indicates a restriction to reduced residue classes. The right-hand side of (24.2) is smaller by a factor (log x)d+1 than the

sum of the corresponding main terms. The proportion of progressions for which the error term in the prime-number theorem for the interval 1 < p G x is greater than (log x) -d times the main term is 0((logx))-1. In calculations we often apply the prime-number theorem to several progressions, and its failure in a small number of cases affects only the error term in the eventual asymptotic formula. An example is the proof that every large even integer is the sum of a prime and an integer with at most three prime factors. It may be that (24.2) holds with some larger constant in place of 2, but 2 is the limit of present methods. An upper bound for the left-hand side of (24.2) is

(

q
1

max li(y, X) I + q
X

H

max ICY, X0) -y I .

P(q) v-,.

(24.4)

ZERO8 AND PRIME NUMBERS

104

4.24

When we use eqn (17.15) to pass to proper characters, the sum in (24.4) does not exceed

+

inaxI0(y,X)I 1-
yq)

v<x

+maxI/(y)-yI L V<X where we have written

,P(q)

+A2Q,

A = log x.

(24.5) (24.6)

The coefficient of the term in X modf in (24.5) is

2 (q) 4=0gdf)

<

Y(f).

Y(f) <x Y(M) C

(24.7)

Relations (16.22) and (24.3) assure us that the second and third terms on the right of (24.5) are xA(24.8) for x sufficiently large. Since f < A4B is of the form (24.1), the SiegelWalfisch inequality (17.14) implies that (24.8) is an upper bound for the terms in (24.5) with f < A4B; we are assuming x is sufficiently large. For q > A4B we combine (17.2) and (17.6) into(24.9)

O(y,X)

where the sum is over zeros of e(s, X) with IpI > R and lyt < T, R and T being chosen for each X so that (17.3) and (17.4) hold; in particular the 0-constant in (24.9) is independent of X, T, or y and -uniform in < R < 1. When we choose T to be a little smaller than x we have

i + 0(x log2x)

AM X) I <

(24.10)

lVt<x

whenever y < x, with an absolute 0-constant. The error term in (24.10) contributes

0(Q x11ogsz)

( 24 . 11 )

to the sum (24.4).

We now divide the rectangle 0 < # < 1, - T < y < T into smaller rectangles, so that /3 is in one of the ranges [0,1+(logx)-1].... , 2 { (r-1)(logx)-1],..., and Iyl in one of the ranges [0, 1], [1, 2],..., [2r, 2r+1],.... Within each rectangle, xf/IpI varies by a bounded factor. Again, we divide the range (MB, Q] for q into intervals

BOMBIERI'S THEOREM

4.24

105

P < q < 2P. We use Ingham's theorem: the number of zeros p = P+iy counted with weight q/p(q) of functions L(s, X) with X proper mod q,

P
(24.12)

(P2U)3(1-a)I(2-a)A6),

G min(P2UA,

by (23.25) and (23.28). Each of these zeros contributes at most a

a

IPI p(q)

q

PU p(q)

(24.13)

to the sum in (24.5). We multiply the expression in (24.12) by Axa/PU and sum over the appropriate sequence of values of PU. First, we have U3(1-a)I(2-a)-1 < A

(24.14)

when a > 2 and the sum is over powers of two not exceeding x. In the sum over P we distinguish two cases. The exponent of P in

(24.12) is greater than unity for a < and less than unity for a > a. The values of P are the powers of two between 2A4B and Q; they are 0(A) in number. Hence for a > 6 we have P6(1-a)I(2-a)-1 / A(A4B)6(1-a)I(2 a)-1

P

< Al+4B(6(1-a)-u (24.15)

x1-aA1-4B,

when we assume x to be sufficiently large. For 2 < a < b we have P6(1-a)!(2-a)-1

a)I(2-a)-1

P x3(1-a)I(2-a)-§A1-B

< 21-aA1-B,

(24.16)

where we have substituted Q= x1A-B from (24.3). The terms from zeros of L-functions formed with characters whose conductors exceed A4B are now seen to be xAlo-B (24.17)

C

by (24.12), (24.13), (24.14), (24.15), and (24.16). The other terms in (24.5) are of the form (24.17), by (24.8) and (24.11), when we choose

B = A+10.

(24.18)

We have thus proved (24.2) with B given by (24.18), which clearly can be improved slightly, since (24.15) and (24.16) can be improved. There are two ways of proving Bombieri's theorem. The shorter, due to Gallagher (1968), is to perform the sieving directly on L'(s, X)/L(s, X)

106

ZEROS AND PRIME NUMBERS

4.24

and related functions in contour integrals. The proof is an anagram of the one we used; multiplication by M(s, x) and fourth-power averages of L(s, x) occur in it. The longer way is to prove a zero-density theorem and deduce Bombieri's theorem from that, as in this chapter. In either case, the Siegel-Walfisch form of the prime-number theorem has to be used, and the constants are non-effective. Without using the hybrid sieve or the approximate functional equation, we should obtain a weaker zero-density result than (23.29), but one still powerful enough to enable (24.2) to be deduced.

25

I. M. VINOGRADOV'S ESTIMATE If we are to capture Baby Roo, we must get a Long Start, because Kanga rims faster than any of Us, even Me. 1. 93

S(a) = S(y,a) = I logpe(pa)

THE sum

(25.1)

ai
is of the type considered in Chapter 6, with local maxima near rational points a/q (if y is sufficiently large). In this chapter, we adapt the proof

of Bombieri's theorem to show rigorously that S(a) is small except possibly when a is close to a rational point with small denominator. More precisely, if B is fixed and x is large, then jS(x,a/q+R)j

<(1+xj/ij)xls-111,

l = logx

where

(25.2) (25.3)

and a/q is in its lowest terms with lB < q G x1-B.

(25.4)

Inequality (25.2) is I. M. Vinogradov's famous `minor arcs estimate', with 8 in place of Vinogradov's z. His proof is elementary, making no use of Dirichlet's series; an analytic proof was published by Linnik (1945).

We remark first that

S(y,a) _ Y_ A(m)e(ma)+0(yilogy), m
(25.5)

the error term arising from squares and higher powers of primes. When a/q is in its lowest terms,

4/ _

A(m)e'-no') fez
\

/`(y, X)+O(logylogq), b nnnQ q

X1no dq CP(b)

(25.6)

the error term arising from powers of the O(log q) primes that divide q; these are not congruent to reduced residues b mod q. The sum of the terms involving b in (25.6) is X(a)T(X), and by eqn (3.20) the modulus

ZEROS AND PRIME NUMBERS

108

4.25

of this expression is at most qi. We combine (24.10) and (17.15) into (25.7)

+ 0(x112)

I0(y, X) I < 1PI Ivy <:C

whenever y < x, the 0-constant being absolute. The sum is over zeros of e(s, Xl), where Xl proper mod f induces X mod q. Since every zero of e(s, Xl) is a zero of L(s, X), we shall take the sum in (25.7) to be over zeros of L(s, X). From (25.5), (25.6), and (25.7), we have I S(y, a/q) I <

1

,p(q)

'Y modq Ipl<>4x I

f Ipl

(25.8)

+ 0(gkx112),

//

where y < x and q < x. We do not use Bombieri's theorem itself, but analogous results for zeros of the functions L(s, X) where X induces a character to the fixed modulus q. This argument is not as natural as that leading to Bombieri's theorem, and the various inequalities do not fit together so well. First we give a form of Ingham's theorem for the set of all characters X mod q, where q is fixed. If X mod q is induced by Xl proper mod f, we have

L(s, X) = L(s, Xl) ]J (1- Xlpp)1.

If

(25.9)

J

The method of Chapter 22 with (19.24) instead of (19.26) gives R(x)

(25.10)

Y* I I L(j+it(r, X)) I4 < pp(f)T log5fT, xmodf r=1 and hence R(x)

IL(J+it(i,, X)) 14 < T log5gT

xmoclq ,

fiq

,P(f)i fJ ('+-) Pi PXf

< T logbgT fJ (p-1+1+p-I) PIq

< qTlog5qT,

(25.11)

since the product over primes dividing q is at most

Here we suppose that the points t(r, X) satisfy conditions (22.23) and (22.24). When we use (18.8) and (25.11) in place of (18.12) and (22.22), Ingham's

theorem becomes: the number of zeros p = l+iy of functions L(s, X) formed with characters X mod q in the rectangle a < (3 < 1, - T < y < T is at most maxipp(q)TA, (o -- .)-§(qT)3(1-a)I(2-ca)(p(q)AO/q)11(2-«)}

where

A = log qT.

(25.12)

(25.13)

4.25

I. M. VINOGRADOV'S ESTIMATE

109

As in the proof of Bombieri's theorem, we sum zeros over rectangles. For a > s the analogue of (24.15) is q3(1-a)I(2-a)-1 < (lB)3(1-a)I(2-a)-1 < x1-al-1B

(25.14)

when x is sufficiently large, and for x < 1 the analogue of (24.16) is q3(1-a)1(2-a)-'j < (xl-B)3(1-a)I(2-a)-} < x1-al-IB.

(25.15)

Since the primes less than q and not dividing it are reduced mod q, we see from the prime-number theorem that y(q)ae/q is greater than unity for q satisfying (25.4). By (25.14), (25.15), and (24.14) the sum over X and p in (25.8) is (25.16) <

and we have proved (25.2) for P = 0. We now show that (25.2) holds (with a different 0-constant) when is different from zero. Since 9M

f 2iii/3e(/3y) dy = e(mnf3)-1,

(25.17)

0

we have the identity S(x, a/q+p)

x

f

= S(x,a/q)+2,ri/3.l

{S(x,a/q)-S(J,a/q)}e(Py) dy.

(25.18)

0

The right-hand side of eqn (25.18) is in modulus < (1+4irIf3Ix)maxjS(y,a/q)I, v;x and we have proved (25.2).

(25.19)

26

I. M. VINOGRADOV'S THREE-PRIMES THEOREM WITH S(y, a) given by eqn (25.1) and x an integer we have 1

f S3(x, a)e(-xa) da = 12 2 logp1 hogp21ogp3,

(26.1)

24 P2 P3

0

where the sum is over triples (p1, p2, p3) of primes with (26.2) P1+p2+Pa = x. We shall find an asymptotic formula for the left-hand side of eqn

(26.1), and deduce that every sufficiently large odd integer is a sum of

three primes. Since we have to use the prime-number theorem for arithmetical progressions, the constant in the error term of the asymptotic formula depends on Siegel's theorem and cannot be stated; however, there are weaker results than Siegel's theorem in which the constants are effective, and in principle a finite set could be given in which any exceptional x must lie. Q =

Let

[x1-23],

(26.3)

where 1 = log x. Since S(x, a) is periodic, we can take the range of integration in (26.1) to be the unit interval [1/(Q+1), (Q+2)/(Q+1)]. We divide this interval into arcs

I, _

[a,.+ar-1 ar+ar+11 q,.+q,.-1' qr+qr+1

(26.4)

corresponding to the fractions a,./q,, of the Farey sequence of order Q. By eqn (9.2) if a,./q,,+p is on I,, then 1P1

< (q, Q)-1

(26.5)

The intervals Ir are known as I'arey arcs (if we think of e(a) as a complex variable, the integration is round the unit circle). We divide the I, into major arcs, on which we work out the integral over the spike at a,./q,, explicitly, as promised in Chapter 6, and minor arcs, on which we

I. iI. VINOGRADOV'S THREE-PRIMES THEOREM

4.26

111

use an upper estimate for the integrand. We call Ir a minor are if q,. > 126,

when (25.2) with B = 26 gives

I<

I S(x,

(26.6)

(1+x1-26Q-1)xl-6 < xl-4

for ar/q,, lying on I, .

On the major arcs we approximate S(a). First we calculate the size of the spike at a/q. From (17.19) we have

loge p51/ p=b(modq)

-[Y]

(26.7)

< x1-32

I

uniformly in q < 126 and in b reduced mod q. We recall that I* e«(ab) = eq(a) = µ(q)

(26.8)

b modq

when a is reduced mod q. Hence

7I

I S(J,alq)-µ(q)[J]Ip(q)I < 1+0Y_gl

iogp AP(q)

p=b(modq)

(26.9)

< p(q)x1-32.

The simplest sum with a spike is

F(a) = f e(moc). By the identity (25.18), S(2,

y(q) F(f3)

q+R)

(26.10)

nacx

<(1+xIflI)naxI S(y,

)

P(q)[y]I

< 128q-1
(26.11)

We now have S3(alq+f3)

3(q) F3(R)

I2+µ2(q) I F(P) l2),

< xl-4I I

1P

\

1P (q)

(26.12)

and the integral in eqn (26.1) is (Q+2)1(Q+1)

f µ3(q,.) (P3(gr ) Trmajor I

f

F3(a-a,./qr)e(-xa) da+0(xl-4

\

r

+0 \ Irmajor

xl-4 y2(q)

J

IS(a)I2 dal +

/

1 /(Q+1)

f

T (q) IrJ

IFaI2da.

q)'

(26.13)

1

The integral in the first error term in (26.13) is

I log2p < xl P<x

(26.14)

ZEROS AND PRIME NUMBERS

112

4.26

by the prime-number theorem or by (8.19), and the integral in the second error term is f IF(p) 12 dfl = X.

(26.15)

Using (8.16) for I µ2(q)/y(q), we see that the error terms in (26.13) are both (26.16) < XI-s.

Finally we must work out the first integral in (26.13). Since e(

F(a)

e( ) a)1 1

we have

eq,(-arx) f

(26.17)

<1

F3(R)e(-x/i) dp-f

g,)e(-xa) daI

F3(oz

i

<

f

(26.18)

M_ < x21-4

lax-

when qr < 126. The integral over /3 on the left of (26.18) is

2x(x-1),

(26.19)

and the integral in eqn (26.1) is now seen to be

2x(x-1) g6128 amodq

oq(

T3(q)

3(q)

+O( 13 2).

(26.20)

We can replace the sum over q G 126 by one over q from 1 to infinity with an error term O(x21-28). Now 00

2* eq(-ax)µ3(q)

2 g°1 amodq

=

lpa(g)

IPs(q)

4

-

2 cZµ(g1d)µ3(q) g

_

T3(q) dlxq

µ2 (q)

dµ(d)

g°6(modd)

cljx

=

µ3(q)eg(-x)

3(q)

(i_

1

\1+(p-/

Ax

H (1+(p±1)3)

pix

((

(p-1)3 1 (p-1)3 (p-1)3+1/ P

12p -3p+3) 1

,

(26.21)

4.26

I. M. VINOGRADOV'S THREE-PRIMES THEOREM

113

which is zero if x is even and positive if x is odd. We put A(x) for the constant in (26.21), so that the left-hand side of (26.1) is 2A(x)x2- O(x21-s),

(26.22)

which is non-zero for x odd and sufficiently large. Since primes less than xl-2 can contribute at most x21-1 to the sum on the right of eqn (26.1), and

l > logp > 1-2logl

(26.23)

if x > p > xl-2, we can assert that the number of solutions of (26.2) is IA(x)x2l-3+ O(x21-4log 1),

(26.24)

an expression which again is seen to be non-zero for x odd and sufficiently

large. The formula (26.24) is the famous theorem of I. M. Vinogradov.

27

HALASZ'S METHOD `And I know it seen easy,' said Piglet to himself, `but it isn't everyone who could do it.'

II. 18

WE have been concerned with estimating how often various sums S are

large. The large-sieve results that we have had gave an upper bound for the sum of 1812 over different values of a parameter. The upper bounds we obtained contained two terms; the first was the maximum of I S 12 and the second the mean square of I S I multiplied by the number

of values of the parameter. Thus if N a(m)rn"d

(27.1)

ma=1

then (18.29) gives L. I S(tr)12 C N log2N G l a(m)12+ (T2-T1)log N I I a(m)12, (27.2) r=1

where the points tr are at least (log N) -1 apart and lie between T1 and T2. The second term in (27.2) corresponds to the mean value of 1S(t) 12 and

the first to its maximum. Halasz's method sometimes enables us to count the number of values of r for which I S(tr) I is large, without the second term's being present on the right of (27.2). Halasz's method is based on the lemma of Chapter 7, R

R n-1

C,.(u,f(r))l

I(f(r),f(q))I2)l. IIuII(I ICrl2)I(15r6Ra=1 max

(27.3)

1

In the proof of (27.3), we assumed that the vectors u, f(1),..., f(R) had finite dimension N, but the proof remains valid when the dimension is infinite, provided that each vector involved has finite norm. We choose the coefficients Cr so that cr(u, f(r)) is real and positive. For (7.15) we took cr to be the complex conjugate of (u, f(r)), but here we give Cr unit modulus,

HALASZ'S METHOD

4.27

115

so that the second factor on the right of (27.3) is R1. If each term the left of (27.3) is at least V we can square (27.3) to give

on

R

R2V2 < RIIuII2 max I I(P)1f(q)) I 1<_rSR q=1

< RIIuII2(maxIlf(r)II2+(R-1)maxl(f(r), f(q))I). r#q l

(27.4)

We can now deduce Halasz's lemma. LEMMA. Let u, f(1),..., f(R) be vectors of finite norm with

I(u,f(r)I > V

(27.5)

R < 2V-2IIu1I2 max Ilf(')II2

(27.6)

for r = 1, 2,..., R. Then

1<-?,-
provided that

Y2 > 2IIu1I2 max I (f(r), f(q)) I.

(27.7)

q#r

We shall apply (27.6) with (u,f(r)) _ rn6N a(m)m-,,

(27.8)

where s,. = ar+itr, 0 < ar < 3. Following Montgomery (1969a), we take urn = e,n!Na(m)

=0

if 1 < m < Nl J' if m > N

fit) = e-mINm-a -itr for all in > 1.

(27.10)

N

Here,

IIu112 < e2 1

and

(27.9)

I a(m) I2

IIf('-)112 < I e-2m1N = IN+O(1).

(27.11)

(27.12)

1

Writing a for ar+aq and t for tq-tr, we have OD

(f(r), f(q)) = I e-2>u/Nm-a-it 1n=1 2+100

=

1

f P(w)(JN)t0 (w+a+it) dw

J

2-i00

by the integral transform (23.6).

(27.13)

ZEROS AND PRIME NUMBERS

118

4.27

Before estimating the integral in eqn (27.13), we move the line of integration to the contour C consisting of C1: the line segment (-ioo, -i(logN)-1], C2: the semicircle, centre the origin, radius (log N)-1, to the right of the imaginary axis, Cg: the line segment [i(log N) -1, ioo).

A residue

(27.14)

1'(1-o-it)(N)i-°-tt

accrues from the pole of (w+a+it) at w+a+it = 1. We recall Stirling's formula in the form (20.3): Ir(A+1T)I =

(27.15)

valid when A < IT j i. Hence if

Itl > logN

(27.16)

the residue (27.14) is bounded.

I. From the approximate fimcNext we need a bound for 10 tionalequation of Chapter 21 we have for 0 < A < 2, IT 1

IS(1'/1-1T)I2 < Y d(ra)ma-1+ In<0 ITI)

1'{J (A+1T+r1Tl)}

+ coax I)I

2 T{J(1-

d(m)+lTl-6 r$o(ITI)

rylt

(27.17)

ITIalOg2ITI,

where we have used (20.4) and (2.14), and partial summation. The functional equation and the estimate (20.4) now give IR(A+iT)12 < ITI1`Alog21T1,

(27.18)

and (27.17) and (27.18) ytogether give IS(A+iT)I <<

ITj1(1-A)loglTJ

(27.19)

uniformly in 0 < A < 1, ITI > 10. We have now shown that (fO, f(2)) I

< 1+ f II'(w) I I (w+A+i') I I JN I w Idw c 00

G 1+ ItlllogltllogN+ f e-17T7T-1(1+IT-tl1log(IT-tl+e)) dT 2

< Itlllog2Nlt1,

(27.20)

4.27

HALASZ'S METHOD

117

where we have used (27.15) for the gamma function and (27.19) for the zeta function. We can now restate the condition (27.8) as T < To, (27.21) where To is such that equality holds in (27.8) when we use (27.20) in

substituting for the scalar product. Hence V2 >

N

Ja(m)I2TOI log2NT.

(27.22)

Clearly when T > To we must divide up the range for T into intervals of length at most To. Repeated application of the inequality (27.7) gives us N R <(To+1)V-2 11Va(m)I2N. (27.23) When we substitute for 1 we have the result which follows: THEOREM. Let

N

G=

If

N I

Ja(m) 12.

(27.24)

rn=1

a(na)m-sl > V

m=1

(27.25)

for s = s1,..., 5R, where s,. = a,.+itr with 0 < yr < J and

T ? It,,-tcl > logN

(27.26)

R < GNV-2+G8NTV-61og4NT,

(27.27)

for q 0 r, then the implied constants being absolute.

The form of the second term in (27.27) arises from our choice of functions f(r); it is larger than the first term unless (27.22) holds with T in place of To. A plausible conjecture is that whenever

R < GNP-2

(27.28)

V2 > GTS

(27.29)

for any fixed 8 > 0. The use of the zeta function to prove (27.27) is a curious feature of Halasz's method. If Lindelbf's hypothesis is true, we can take the line of integration in (27.13) to Re w = J, with the effect of replacing To in (27.22) by N=To for any e > 0. This is an improvement for T > N (and if T < N then (27.28) follows trivially from (27.2)), but it is still a long way from weakening the condition on To to (27.29).

28

GAPS BETWEEN PRIME NUMBERS `I shall do it', said Pooh, after waiting a little longer, `by means of a trap. And it must be a Cunning Trap, so you will have to help me, Piglet.' I. 56

F I P. S T we prove a theorem on the zeros of c(s), replacing the large sieve

(19.26) by Halasz's method in the work of Chapter 23. We shall use the notation of that chapter with Q = 1, so that only zeros of the zeta function are considered. The definition of class (i) and class (ii) zeros remains as before. We pick representatives of each class of zeros in such a way that their imaginary parts differ by at least 21, where

l = log T,

(28.1)

but the representatives are in number > l-2 times the zeros in that class.

We suppose a > J, since the result (28.19) which we obtain below improves on Ingham's theorem only for a > 1. The parameters X and Y will satisfy X < T2, 1001Y < T2. (28.2) In the definition (23.14) of a class (ii) zero p = f3+iy, 1-S+ioo

f

0P+w)M(p+w)Y"F(w) dwl > 17r,

(28.3)

f-5-100

the parts of the integrand with 1Im w l > 1001 give less than I (if 1 is sufficiently large). The integral of IP(j+it) I converges rapidly so, for (28.3) to hold, there must be some t with It-yl < 1001 for which N+it)M(- -+it) I > cYR-4,

(28.4)

where c is an absolute constant. We pick as representatives of the class (ii) zeros a sequence of values of t satisfying (28.4). By (22.22) the number of these t with

I*+it)I > U,

(28.5)

where we choose U below, is T U-415.

(28.6)

GAPS BETWEEN PRIME NUMBERS

4.28

119

Otherwise we have I111(j+it) I > V = cU-1Y01-',

(28.7)

and by (27.28) the number of such t is XV-21+XTV-617. We choose

(28.8)

U = X_111oy3(2a-1)110

V=

(28.9) (28.10)

cX1110Y(2a-1)15,

and on adding (28.6) and (28.8) and multiplying by 12 we see that class (ii)

zeros number

< Xr216y-6(2a-1)/5T19T _ I X'416Y-2(2a-1)/613,

(28.11)

the second term in (28.11) being less than the first provided (28.12)

X2Y4(2a-1) << T5130.

A zero is of class (i, r) if (28.13)

Y a(m)m-Pe-1111r I > {20(r2+1)}-1. ME1,

We pick representatives and apply (27.28) with

G= G 17LEjr

(28.14)

Ia(m)I2n2-tae-2m/Y < (2rY)1-20exp(-21')l3.

The number of representatives is thus r4(2'Y)2-2aexp(_ 2r+1)13+rl2(2rY)4-8aT exp(- 3 .

2r+1)113.

(28.15)

Summing over r and multiplying by l2, we see that there are at most <<

Y2-2a19+X4-6aTl27

(28.16)

class (i) zeros. Choosing X = Tk(2a-1)1(a8+a-1)

(28.17)

Y - T1(sa-3)1(aa+01-1)

(28.18)

we find (28.12) is satisfied for 0 < a < 1, and that the number N(a, T) of zeros p = /3-]-iy of C(s) with /3 > a and l y I < T satisfies the relation

N(a, T) <

T((5(x-3)(1-a))1(a2 +a-1)127

for < a < 1. The result (28.19) is also true for

(28.19)

a < I by Ingham's

theorem.

We now sketch the proof of our theorem on gaps between prime numbers. THEOREM. Let c be a real number greater than i22. Then whenever x is sufficiently large, there is a prime p with

x < p < x+xc. 853618 X

I

(28.20)

120

ZEROS AND PRIME NUMBERS

4.28

Such a result was first proved by Hoheisel (1930) with c a little less than one. Ingham (1937) obtained the result with c > a and indicated how to replace - with a smaller number by improving an upper bound for Several authors achieved this by means of intricate arguments. Recently Montgomery (1969b) obtained the result for c > L

by the method given here, but with a less efficient use of the Halasz lemma; the improvement to i 2 was seen by the author in preparing the present exposition. As we have seen, Montgomery's method rests on the Halasz lemma, and thus on bounds for I (1 +it) 1. As with Ingham's

result c > -, improvements at I+ it improve the constant, in that a good estimate for the mean of a higher power than I (++it) I4 would decrease the estimate for class (ii) zeros, both in (28.19) and in Ingham's theorem. However, even if we knew Lindelof's hypothesis, we should only be able to deduce (28.20) for c > 1. It has long been conjectured (Cramer 1936) that for large x there is always a prime p with

x < p < x+0(log2x),

(28.21)

but there seems no chance of approaching this conjecture by present methods.

There are two essentials for a proof of (28.20) with c < 1: a zerodensity theorem such as (28.19) and a result on zeros of c(s) with fi close

to 1. We shall assume that c(s) has no zeros p = fl+iy with p > 1-A{log(IYI+e)}-B,

(28.22)

where B < 1. The inequality (28.22) is proved by Hadamard's doubleheight method just as (13.12) was, but the proof uses such inequalities as

IW+it)I <10,0Itl+e)

(28.23)

with c < 1. No better way to prove bounds for Ig(1+it) I is known than to replace exp(-itlogm) by exp{-itP(m)}, where P(m) is a polynomial arising from the first few terms in the expansion of the logarithmic series. Since m runs through integer values, the resulting sum depends only on the fractional part of 2t/v. More direct arguments fail, because t is much larger than any other parameter involved. After this transformation we must use the intricate methods of I. M. Vinogradov; Weyl's simpler approach gives only I C(J+it) I< log( I t I +e)/loglog(I t I +e2),

(28.24)

where any higher power of loglog t than the first would suffice for the application. The proof of (28.22) and (28.23) occupies one and a half chapters of Titchmarsh (1951).

4.28

GAPS BETWEEN PRIME NUMBERS

121

While proving the prime number theorem in Chapter 16 we saw that

xP+

Oll/x log2xl

Iyl
(28.25)

T J'

where T is chosen less than x with the property that each zero p = f+iy of c(s) has (28.26) ly-TI . log T, the sum in (28.25) being over all zeros p of e(s, X) with lye < T. Hence

XP-(- h)P+O(Y) 2

qi(x+h)-fi(x) = h+

(

28.27)

IYI
where we have written

(28.28)

A = log x.

By the mean-value theorem the sum over zeros in (28.27) is in modulus

I h(x+Bh)P-1 < h IyI
Iyl
0-1

(28.29)

for some 0 in 0 < 0 < 1. To estimate the sum in (28.29) we divide the interval [0, 1] into ranges [0, J],

The number of zeros p = P+iy with a < < a+A-1 is << T15 (1-a)A27,

(28.30)

this being by (28.19) for a > j and by Ingham's theorem (23.29) for a < 1. The sum in (28.29) is thus << hA28 max

(xT-1215)S-1

(28.31)

lyl
the maximum being over zeros p with lyI < T. With

T=

x6112-5,

(28.32)

where 8 > 0, the expression in (28.31) is ha28 max x80-1) < hA28 exp{8A(AA-B)},

(28.33)

lyl
which is o (1) as x tends to infinity; here we have used (28.22). The error term in (28.27) is also less than h when h > x7112+8A2

(28.34)

If (28.34) holds with a sufficiently large constant then

O(x+h)- &(x) > {I-o(1)}h.

(28.35)

122

ZEROS AND PRIME NUMBERS

4.28

Finally we note that the prime powers up to x+h contribute x1A2

to the sum (x+h)-fi(x), and so for sufficiently large x Y log p > 1h,

(28.36)

(28.37)

x
and we have proved (28.20) when we choose 12

(28.38)

NOTATION 1, II

indicate a sum or a product over primes only (Chapter 1)

D

D

(m, n)

highest common factor of the integers in and n (Chapter 1)

m - n (modq) m-n is a multiple of q (Chapter 1) e(a) = exp27ria, eq(a) = exp(27ria/q) (1.4)

1

sum over a set of representatives of residue classes mod q

a mod q

(Chapter 1) sum over a set of representatives of reduced residue classes mod q (Chapter 1) Euler's function (1.10)

amodq

cp(m)

d(m) µ(rn) A(m)

Ramanujan's sum (1.11) complex variable (1.16) a Dirichlet's character (Chapter 1) the number of divisors of in (1.23) MSbius's function (1.25) log p if in is a prime power pa, otherwise 0 (1.31)

O(x)

sum ftmmction of A(m) (2.1)

f(x) <
Win)

If(x)I = o(g(x)) (2.3) the conductor of a character (Chapter 3) Gauss's stun (3.7) a trivial character (Chapter 3)

[a]

largest integer not exceeding a (4.3)

cq(m)

s = v+it X(rn)

f (X) T(X)

11.11

H(a) L(s, X) c(s)

S(a) rr(x)

a/q T(s) 6(8), e(s, X)

distance of a from the nearest integer (4.4) the saw-tooth Fotuier series (4.5) Dirichlet's L-ftmction (5.8) Riemaim's zeta function (5.9) an exponential sum (Chapter 6) number of prunes up to x (Chapter 6) a rational number in its lowest terms (Chapter 6) Euler's gamma function (11.1), (11.12) functions occurring in function equations for c(s) and L(s, X) (12.2), (12.3)

p = P+iy Ox, X) O(x; q, a)

a zero of e(s) or of 6(s, X) (Chapter 12) sum ftmction of A(m)x(m) (17.1)

sum ftmction of A(m) in the arithmetic progression a (mod q) (17.18)

SX

Y_*

xmodq

u = A-I-iT G(u) M(s, X) N(X)

a character sum corresponding to the exponential stun S((X) (18.4)

a stun over proper characters modq (Chapter 18) auxiliary complex variable (Chapter 20) all the `junk' in the functional equation (20.2) partial stun for the inverse of L(s, X) (23.2) number of zeros of L(s, X) in a rectangle (Chapter 23)

BIBLIOGRAPHY The epigraphs are from I. TYinnie the Pooh H. The House at Pooh Corner

BODIBIERI, E. (1965). On the large sieve. Mathematika 12, 201-25.

-

- (1972). A note on the large sieve. To appear. - and DAVENPORT, H. (1968). On the large sieve method. Abhandlungen aus Zahlentheorie and Analysis zur Erinnerung an Edmund Landau. Berlin. (1969). Some inequalities involving trigonometric polynomials. Annali Scu. norm. sup., Pisa 23, part 2, 223-41. CHANDRASEKHARAN, K. and NARASIMHAN, R. (1963). The approximate functional

equation for a class of zeta-functions. 11lath. Annaln 152, 30-64. CRAA R, H. (1936). On the order of magnitude of the difference between consecutive prime numbers. Acta arith. 2, 23-46. DAVENPORT, H. (1967). lllultiplicative number theory. Markham, Chicago. ESTERMANN, T. (1948). On Dirichlet's L-functions. J. Lond. math. Soc. 23, 275-9. FOGELS, E. (1969). Approximate functional equation for Hecke's L-functions of

quadratic field. Acta arith. 16, 161-78. FRANEL, J. (1924). Les suites de Farey et lea problbines des nombres premiers. Nachr. Ge8. TPiss. Gottingen, 198-201. GALLAGHER, P. Y. (1967). The large sieve. 111athematika 14, 14-20.

(1968). Bombieri's mean value theorem. Ibid. 15, 1-6. HALAsz, G. and TuRAN, P. (1969). On the distribution of the roots of Riemann Zeta and allied functions I. J. Number Theory 1, 121-37. HALBERSTAM, H. and ROTH, K. F. (1966). Sequences, vol. 1. Oxford. HARDY, G. H. and AVRIGHT, E. M. (1960). An introduction to the theory of numbers.

4th edn. Oxford. HOHEISEL, G. (1930). Primzahlprobleme in der Analysis. Sber. berl. math. Ges. 580-8. INGHAM, A. E. (1937). On the difference between consecutive primes. Q. Jl 111ath. 8, 255-66.

- (1940). On the estimation of N(a, T). Ibid. 11, 291-2. JEFFREYs, H. and JEFFREYS, B. (1962). Methods of mathematical physics. Cambridge. LANDAU, E. (1927). Porlesungen ilber Zahlentheorie. Leipzig.

LrzNIIx, Yu. V. (1945). On the possibility of a unique method in certain problems of additive and multiplicative number theory. Doklady Akad. 1\Tauk SSSR, ser. mat. 49, 3-7. - (1964). All large numbers are sums of a prime and two squares (a problem of Hardy and Littlewood) II. Am. math. Soc. Transl. (2) 37, 197-240. and RANYI, A. (1947). On some hypotheses in the theory of Dirichlet characters. Izv. Akad. Nauk SSSR, ser. mat. 11, 539-46. MONTGOMERY, H. L. (1968). A note on the large sieve. J. Lond. math. Soc. 43, 93-8.

(1969a). Mean and large values of Dirichlet polynomials. Invent. math. 8, 334-45.

- (1969b). Zeros of L-functions. Ibid. 8, 346-54. - (1971). Lectures on multiplicative number theory. Springer.

BIBLIOGRAPHY

125

PBLYA, G. (1918). Uber die Verteilung der quadratisohen Reste and Nichtreste. Nachr. Gee. Wise. Gottingen, 21-9. PRACHAR, K. (1957). Primzahlverteilung. Springer. ROTH, K. F. (1965). On the large sieves of Linnik and R6nyi. Mathematika 12, 1-9. TITCHarARSH, E. C. (1951). The theory of the Riemann zeta-function. Oxford. VINOGRADov, A. I. (1965). On the density hypothesis for Dirichlet L-functions.

Izv. Akad. Nauk SSSR, ser. mat. 29, 903-34. VINOGRADOV, I. M. (1954). The method of trigonometric sums in the theory of

numbers (transl. A. Davenport and K. F. Roth). Interscience, New York. - (1955). An introduction to the theory of numbers (translation). Pergamon, Oxford.

INDEX abscissa of covergenoe, 18 additive fimction, 5

Gauss, 11, 22 Gauss's sum, 11-13

approximate functional equation, 84-93, 106, 116

arithmetical functions, 2 additive, 5 multiplicative, 2 totally multiplicative, 2

Hadamard's product, 50-4, 69 Hal>sz's method, 30, 114-18 ineffective constants, 62, 72, 110 Ingham's theorem, 98-102, 105,

integral functions, 50, 85

characters: conductor of, 11

Jensen's formula, 51-2

defined, 4 induced, 11 propriety of, 11 sieved, 74-6, 79-82 trivial, 3, 13 classification of zeros, 99-100 congruence 1 Dirichlet :

characters, see characters divisor problem, 8 polynomials, 80, 84 series, defined, 18 divisor function : averaged, 7-9 defined, 4 elementary proofs, 69, 107 Euler's : function defined, 3 product, 20, 52 exceptional zero, 60-5, 70-72, 76 exponential maps, 1 exponential sum : defined, 24 sieved, 28-31, 77 spikes of, 24-6, 110-11 Farey : arcs, 110 sequence, 36, 74, 110 Fourier : series, 14, 37, 40-1 transform, 79

Franel's theorem, 36-9, 74

functional equation, 20, 41, 45-50, 69, 84, 116

Gallagher's :

first lemma, 76-8, 81-2 second lemma, 80

108,

118-19, 121

Bombieri's theorem, 35, 103-9

L-functions defined, 19-20 Lindelof hypothesis, 93, 117, 120

major arcs, 110-12 minor arcs, 107, 110-11 modular : functions, 43 relation, 43 M6bius :

function defined, 4 inversion, 5, 33, 38 multiplicative function defined, 2

Plancherel's identity, 79-80 Poisson's summation formula, 40-42 Polya's theorem, 14-17, 50, 63 prime number theorem :

for an arithmetic progression, 23, 62, 71-2, 74, 103, 110 forms of, 21-2, 23, 39, 55, 62 proved, 66-72 propriety of characters, 11

Ramanujan's sum, 3, 13, 24, 26, 38, 111 representative zeros, 100, 119 residues :

classes of, 1-2 complete set of, 2, 10-11 reduced, 3, 10 Riemann hypothesis, 20, 52

Schinzel's hypothesis, 23-5 Selberg, A., 21-2, 26-7 Siegel's theorem, 62-5, 71-2, 110 Siegel-Walfisch theorem, 72, 104, 106 sieve :

for characters, 74-6 for exponential sums, 28-31, 77 hybrid, 79-83, 93, 101

128

INDEX

sieve (coat.) large, 27, 28-35, 38, 74-8, 114, 118

upper bound, 25-7, 32-5 sifted sequence, 25, 32-4 spikes of an exponential sum, 24-6, 110-11

Vinogradov, I. M., 8, 14, 25, 57, 107, 113. 120

estimate, 107-9 method, 57, 120 theorem, 113

Stirling's formula, 47, 51, 52, 84, 99, 116 totally multiplicative functions, 2 trigoncmetric sums, see exponential sums uniform distribution, 73-4 Vinogradov, A. I., 98-9, 103

Weyl's criterion, 73 zero-density theorems, 98-102, 105, 108-9, 118-20 zero-free region, 56-65, 120 zeta-function defined, 19-20