simple zeros of the riemann zeta-function · 2021. 1. 14. · simple zeros of the riemann...

SIMPLE ZEROS OF THE RIEMANN ZETA-FUNCTION

J. B. CONREY, A. GHOSH and S. M. GONEK

[Received 17 July 1995ÐRevised 1 April 1997]

1. Introduction and notation

Let zs denote the Riemann zeta-function and r b ig a zero of zs. In theprocess of evaluating NT , the number of zeros of z with 0 < g < T , each zero iscounted with multiplicity, denoted by mr, to give the von Mangoldt formulae

NT T2p

logT

2pe 7

8 ST O 1

T

� �;

ST 1p

arg z12 iT p log T :

It is not known if mr 1 for all r but it is so conjectured. Indeed, all the zeroslocated so far, more than 1:5 ´ 109 of them, are simple. Support for thisconjecture is further strengthened by the fact that Montgomery's pair correlationconjecture (see [12]) implies that almost all the zeros are simple.

It is known that at least two-®fths of the zeros are simple. (See Conrey [3, 4]for a discussion of results in this direction). Assuming the Riemann Hypothesis(RH), Montgomery [12] has shown that at least two-thirds of the zeros are simple,and later, with Taylor [13], the constant two-thirds was improved to

3

2ÿ

2p

2cot

2p

2 0:6725 . . . :

Cheer and Goldston [1] have recently improved upon this result slightly byshowing that RH implies that at least 0.672753 of the zeros are simple.

A problem related to the simplicity of the zeros of zs is the enumeration ofdistinct zeros (that is, to count each zero precisely once without regard toits multiplicity).

Let us de®ne

NdT jfr b ig: 0 < g < T ; zr 0gj;NsT jfr b ig: 0 < g < T ; zr 0; mr 1gj:

We are interested in obtaining bounds for the constants Cd and Cs de®ned by

Cd lim infT!1 NdT =NT ;

Cs lim infT!1 NsT =NT :

This research was supported in part by an NSF grant.

1991 Mathematics Subject Classi®cation: 11M26.

Proc. London Math. Soc. (3) 76 (1998) 497±522.

Of course, it is clear that Cd 1 if and only if Cs 1. To get a bound for Cs ,Montgomery proceeded by observing that

NsT >Xg < T

2ÿ mg 2NT ÿXg < T

mg

(where all sums over zeros are repeated according to multiplicity and we usemg in place of mr on assuming RH), and he showed in [12] thatX

g < T

mg< C0 o1NT C0 43 ; T ! 1:

One gets a bound for Cd by observing that

2NsT <Xg < T

mg ÿ 2mg ÿ 3mg

Xg < T

mg ÿ 5NT 6NdT ;

so that

Cd >165 2Cs ÿ C0: 1:1

Due to the relationship between C0 and Cs in Montgomery's method, we have

Cd >169ÿ 3C0> 56 if C0 43 : 1:2

(The constants are improved very slightly by using the better value for C0 in [13]or in [1].) If one could devise a method for evaluating Cs independently of C0,then (1.1) would be essentially better than (1.2).

In this paper we develop a new method to obtain lower bounds for Cs and Cd.Our method requires the assumption of the Riemann Hypothesis as well as anassumption of an upper bound for averages of sixth moments of Dirichlet L-functions Ls; x. This sixth moment hypothesis is implied by the GeneralizedLindeloÈf Hypothesis, which is the assumption that for any « > 0,

Ls;xp« q1 j t j«

for j > 12, where x is a character modulo q. The Generalized LindeloÈf Hypothesis

is weaker than the Generalized Riemann Hypothesis which conjectures thatLs; x 6 0 if j > 1

2. For convenience, we state our main result as follows.

Theorem 1. Assuming the Riemann Hypothesis (RH) and the GeneralizedLindeloÈf Hypothesis (GLH), we have

Cs >1927

; Cd >56 1

81:

Remark. In an earlier version of this paper, we had omitted the assumption ofGLH. We thank the referee for pointing out this mistake. Our paper [5], whichrefers to the methods here, requires this additional assumption of GLH as well.Also, we would like to thank especially a second referee who gave a considerablesimpli®cation for the evaluation of our main terms in § 8. We have included thissimpli®cation here.

498 j. b. conrey, a. ghosh and s. m. gonek

1.1. NotationWe shall use � to denote Dirichlet convolution of arithmetic functions. Thus if

a and b are arithmetic functions, then

a � bn Xd jn

ad b nd

� �:

Further, we de®ne operators Ts and Lk by

Ts an nsan;

Lkbn log nkbn:Note that

Tsa � b Ts a � Ts band

La � b La � b a � Lb:Also, observe that if k is squarefree and a is multiplicative (that is, amn aman wherever m; n 1), then

ab � gk ab � agk:We use tr to denote the r-fold divisor function (often denoted dr ). Often this

has an unspeci®ed but bounded positive integer subscript r, not necessarily thesame at each occurrence. We make frequent use of the inequalities trntsn<trsn and trmn< trmtrn for positive integers r, s, m and n. We use I forthe arithmetic function which is the identity for Dirichlet convolution, that is,I1 1 and In 0 for n > 1. Also, we use 1 for the arithmetic function forwhich 1n 1 for all n.

Throughout L shall denote logT =2p, and ex denotes e2pix.

2. Sketch of the proof for Cs

We note that r is a simple zero of zs if and only if z 0r 6 0. By Cauchy'sinequality, it follows thatX

g < T

Bz 0r�� 2

and

bk mkP log y=klog y

� �; 2:4

where Px is a polynomial with real coef®cients which satis®esP0 0; P1 1:

Then, if v < 12, we shall prove the following theorem.

Theorem 2. Assuming GLH, we have

S1 :Xg < T

Bz 0r, 12 v

Z 10

Px dx� �

TL2

2p2:5

and

S2 :Xg < T

Bz 0rBz 01ÿ r, l TL3

2p; 2:6

where

l 13 v

Z 10

Px dx� �2

vZ 1

0Px dx 1

12v

Z 10

P 0x2 dx: 2:7

Assuming the Riemann Hypothesis, we have S2 P

g < T jBz 0rj2. (This is theonly place we need RH.) Our estimate for Cs is then obtained by choosing thepolynomial Px optimally. By the calculus of variations, the optimal choice is

Px ÿvx2 1 vx:Then Z 1

0Px dx 1

2 1

6v and

Z 10

P 0x2 dx 13v2 1;

so that with v 12ÿ « «! 0 we obtain

S1 , 1924 NT L and S2 , 5764 NT L2:Theorem 1 then follows.

To obtain the formulae (2.5) and (2.6), we use Cauchy's residue theorem towrite

S1 1

2pi

ZC

z 0

zsz 0sBs ds 2:8

and

S2 1

2pi

ZC

z 0

zsz 0sz 01ÿ sBsB1ÿ s ds; 2:9

where C is the positively oriented rectangle with vertices at 1ÿ c i, c i,c iT , and 1ÿ c iT ,

c 1Lÿ1; 2:10


and T is chosen so that the distance from T to the nearest g is qLÿ1. (It is easyto see that this assumption on T involves no loss of generality.)

Remark. There is a fair amount of work involved in evaluating S1 and S2,even if one considers only main terms without regard to error terms. Therefore,let us motivate this work by a suggestion as to what we might expect, byconsidering averages over all t instead of just ordinates of zeros of zs. In thiscase we would start with the function GsBs instead of z 0sBs, where (see(3.2) for the de®nition of xs)

Gs z 0s ÿ x0

xszs

and for P we take Px x. Note that G and z 0 agree at s r. Then it is trivialto see that Z T

1GB1

2 it dt , TL

by writing the integral as a complex integral and moving the line of integration toj c > 1. Also, we know thatZ T

1jGB1

2 itj2 dt , 4

3TL2

since Levinson evaluates this integral in his work [10]. (He considers the integralmore generally on the a-line where 1

2ÿ a p log T ÿ1.) Thus, consideration of

integral means leads one to suspect that our method might lead to a proof thatthree quarters of the zeros are simple. However, one can see from the formulae(2.5), (2.6), and (2.7) that the average at zeros is slightly different from theaverage over all t.

3. Preliminary transformations

The ensuing analysis is much simpler if, instead of (2.8) and (2.9), we useslightly different expressions.

The functional equation for zs is given byzs xsz1ÿ s; 3:1

where

x1ÿ s xsÿ1 22pÿsGs cos 12ps: 3:2

We differentiate to obtain

z 0s ÿxs z 01ÿ s ÿ x0

xsz1ÿ s

� �: 3:3

We ®rst consider S1 . It follows from (2.5) and (3.3) that

S1 ÿXg < T

xrz 01ÿ rBr

12pi

ZC

z 0

z1ÿ sxsz 01ÿ sBs ds: 3:4

501simple zeros of the riemann zeta-function

Let s j it be on C. We shall use the following well-known estimates:xsp j t j1=2ÿj; 3:5Bsp y1ÿjL; 3:6

z 01ÿ sp j t jj=2L2; 3:7and

z 0

z1ÿ spL2 3:8

for s on C; the last estimate holds if the distance from t to the nearest g isqLÿ1 which we have assumed. Hence, the contribution to S1 from the integral in(3.4) taken along the horizontal sides of C is

pT c=2ycL5 p T 1=2yL5: 3:9To estimate the contribution to S1 from the right-hand side of the contour C,

we use (3.3) to replace xsz 01ÿ s in (3.4). Also, by (3.1) and (3.2),z 0

zs z

0

z1ÿ s x

0

xs ÿ log j t j

2p O 1

1 j t j� �

: 3:10

Thus, the integral along the right-hand side of C is equal to

1

2pi

Z ciTci

x 0

xs2zs ÿ 2 x

0

xsz 0s z

0

zsz 0s

� �Bs ds

T2pL2 OTL; 3:11

since only the term n 1 from the Dirichlet series for zsBs contributesanything to the main term.

To evaluate the contribution to S1 from the left-hand side of the contour C, wemake the change of variable s! 1ÿ s in (3.4). Then, the integral along the left-hand side is equal to ÿÅI1, where

I1 1

2pi

Z ciTci

x1ÿ s z0

zsz 0sB1ÿ s ds: 3:12

So far, we have shown by (3.4), (3.9), (3.11), and (3.12) that

S1 T

2pL2 ÿ ÅI1 OTL OT 1=2yL5: 3:13

We now consider S2 . By (2.6), (3.3), and Cauchy's theorem,

S2 ÿXg < T

x1ÿ rz 0r2B1ÿ rBr

ÿ 12pi

ZC

x1ÿ s z0

zsz 0s2BsB1ÿ s ds: 3:14

By the estimates (3.5)±(3.8), the contribution to S2 of the integral over thehorizontal sides of the contour is

pT 1=2yL8: 3:15


The integral along the left-hand side of C is, by (3.3) and (3.10), equal to

1

2pi

Z 1ÿciT1ÿci

x 0

xs ÿ z

0

z1ÿ s

� �xs x

0

xsz1ÿ s ÿ z 01ÿ s

� �2BsB1ÿ s ds;

which, by the change of variable s! 1ÿ s and another use of (3.3), is

ÿ 12pi

Z cÿiTcÿi

x1ÿ s

´x 0

xs3zs2 ÿ 3 x

0

xsz 01ÿ sxs ÿ z

0

zsz 0s2

� �BsB1ÿ s ds: 3:16

We multiply this out and rewrite it as a sum of three integrals. For the ®rst two ofthese we move the line of integration to the line j 1

2with the same error term

as in (3.15). Thus the above is equal to

M1 M2 ÿ ÅI2 OT 1=2yL8; 3:17where

M1 1

2p

Z T1

x 0

x1

2ÿ it3jBz1

2 itj2 dt; 3:18

M2 ÿ3

2p

Z T1

x 0

x1

2ÿ itjBz 01

2 it j2 dt; 3:19

and

I2 1

2pi

Z ciTci

x1ÿ s z0

zsz 0s2BsB1ÿ s ds: 3:20

The integral in (3.14) taken along the right-hand side of C is equal to ÿI2. Thus,by (3.15) and (3.17), we see that

S2 M1 M2 ÿ 2RI2 OT 1=2yL8: 3:21The integrals in M1 and M2 have been evaluated (implicitly) by Conrey in [2].Let

Fs XNn0ÿ1nan znsLÿn

X1m1

Qlog m

L

� �mÿs; 3:22

where

Qx XNn0

an xn

is a polynomial with real coef®cients. Let

M Z TU

TjBF1

2 itj2 dt; 3:23

where

U TLÿ10


and Bs is as in (2.2). Then

M U Q02 vZ 1

0

Z 10

PuQ 0v 1v

P 0uQv� �2

du dv

� � OULÿ1logL5; 3:24

provided v < 12. Moreover, for T < t < T U , we have

x 0

x1

2ÿ it ÿL OLÿ10: 3:25

Thus, we can replace x 0=x by ÿL and sum integrals of length U to obtainestimates for M1 and M2. For M1 we take Q1x 1, and for M2 we takeQ2x ÿx. We then obtain

M1 ÿTL3

2p1 1

v

Z 10

P0u2 du� �

OTL2logL5 3:26and

M2 3TL3

2p12 v

Z 10

Pu2 du 13v

Z 10

P 0u2 du� �

OTL2logL5: 3:27

We have shown that (3.26) and (3.27) are valid for v < 12

as T ! 1. It is worthremarking that (3.26) and (3.27) are valid for v < 4

7, by the work in [4].

It remains to estimate I1 and I2 in (3.12) and (3.20). Indeed, this forms themain area of dif®culty and requires a bit of preparation, in the form of lemmas,which we state in the next section.

4. Preliminary lemmas

Lemma 1. Let r > 0. Then, for any c0 > 0,

1

2pi

Z ciTci

x1ÿ srÿs ds eÿ2pir Er; crÿc if r < T=2p;

Er; crÿc otherwise;

(4:1

uniformly for c0 < c < 2, where

Er; cp T cÿ1=2 Tc1=2

jT ÿ 2prj T 1=2 : 4:2

This is provided implicitly by Gonek in [8, Lemma 2].

Lemma 2. Suppose that As P1n1 annÿs for j > 1, whereanp tk1nlog nl1

for some non-negative integers k1 and l1. Let Bs P

n < y bnnÿs, wherebnp tk2nlog nl2

for non-negative integers k2 and l2 and where

T e p y p T


for some e > 0. Also, let

I 12pi

Z ciTci

x1ÿ sB1ÿ sAs ds;

where c 1 1= log T. Then we have

I Xn < y

bnn

Xm < nT =2p

ameÿm=n OT 1=2ylog T A

for some A.

Remark. An admissible value for A is k1 k2 l1 l2.

Proof. By Lemma 1 it suf®ces to show thatXn < y

jbnjX

m

jamjmc

T 1=2 fm;nT p yT 1=2log T A;

where

fm;nT T 3=2

jT ÿ 2pm=nj T 1=2 :

Now by a theorem of Shiu [15], for any e > 0,XXÿY < n < X

tknlog nl pe Ylog X klÿ1 4:3

provided that Y q X e. It follows easily thatXn < y

jbnjX

m

jamjmc

p ylog T A

with A k1 k2 l1 l2 ÿ 1. Thus, the term with the T 1=2 is acceptable. Next,we break up the sum involving fm;n into three parts. The terms withjT ÿ 2pm=nj > 1

2T have fm;nT p T 1=2, so this case is like the one we have just

discussed. Now consider the terms for which

T 1=2 < jT ÿ 2pm=nj< 12T :

Assume ®rst that

T 1=2 < 2pm=nÿ T < 12T ;

the other inequality leading to a similar argument. We further re®ne the sum intop log T sums of the shape

T P < 2pm=n < T 2Pwhere T 1=2 p P p T . For m and n satisfying this condition, fm;nT p Pÿ1. Also,m < nT and m ranges over an interval of length

which, by Shiu's theorem, is

p yT 1=2log T A

for some A. Summing this over the p log T values of P contributes an additionalfactor of log T .

Finally, we consider the contribution of the terms for which

jT ÿ 2pm=nj< T 1=2:For such values of m and n we have fm;nT p T ÿ1=2. For a given n theadmissible values of m are 1,


where

janj< Ktknlog nl:Then for any « > 0 we have

R :Xn < x

an ÿ 12pi

Z ciUcÿiU

As xs

sds p« Kx

«1 xUÿ1;

where c 1 log xÿ1 and the implied constant is independent of K. IfU p x1ÿ« for some « > 0, then

R p« KxUÿ1log xkl:

Proof. By the lemma of § 17 of Davenport [6],

R pX1n1n 6 x

x

n

� �c janjmin1; Uÿ1j log x=njÿ1 jaxj;the last term occurring only if x is an integer. The lemma follows easily fromthis, (4.3), and the fact that anp« n« for any « > 0.

Lemma 5. Suppose that y, d p T with log y q log T. Let

Sd; M :X

k < y=d

bkd klog kM :

Then for any « > 0,

Sd; M ÿ1M md d

fd P1ÿM log y=d

log y

� �log yMÿ1 O«EM if 0 < M < 1;

O«EM if M > 2;

8 0 and

F1d Yp jd1 pÿ3=4:

In all cases,

Sd; M p« LMÿ1«F1d :

This follows from Lemma 10 of Conrey [2].

Lemma 6. If Q is a polynomial, thenXd < y

m2d fd Q

log y=d

log y

� �

Z 10

Qu du� �

log y O1:

This is well-known in the case Qx � 1 and the general case may be deducedfrom this particular one.


Lemma 7. Suppose that Q is a squarefree positive integer. Then for boundedr > 0 and ®xed j, with 0 < j < 1, we haveX

d jQtrd dÿj p expClog Q1ÿjlog log Qÿ1

and Xd jQ

trd dÿ1 p log log QC

for some positive constant C which is independent of Q.

Proof. Since the summand is multiplicative, we have

logXd jQ

trd dÿj Xp jQ

log1 tr ppÿjpXp jQ

pÿj:

If the primes p jQ are denoted p1; . . . ; pn and the ®rst n primes are q1; . . . ; qn, thenqn < 2 log Q if Q is large enough. Moreover, p

ÿji < q

ÿji so that, by the Prime

Number Theorem, the above sum is

pX

q < 2 log Q

qÿj pZ 2 log Q

2ÿuÿj dpu

plog Q1ÿjlog log Qÿ1 if 0 < j < 1;log log log Q if j 1:

(The result now follows.

5. Estimation of I1 and I2

By (3.12) and (3.20), we have

In 1

2pi

Z ciTci

x1ÿ sAnsB1ÿ s ds; 5:1

where n 1 and 2, with

A1s z 0

zsz 0s

X11

a1mmÿs 5:2

and

A2s z 0

zsz 0s2Bs

X11

a2mmÿs: 5:3

Thus, by Lemma 2 with k 4 and l 3, we get In Mn En where

Mn Xk < y

Xm < kT =2p

anmbkk

e ÿmk

� �5:4

and

En p T1=2yL7: 5:5


To evaluate Mn we apply Perron's formula to the sum over m and the mainterm arises from the residue of the pole at s 1 of the generating function. If weassume the Generalized Riemann Hypothesis, then the error terms are easilyhandled. For our less conditional treatment, however, we must use Perron'sformula at a later stage after some preliminary rearrangements of the sum.

We ®rst express the additive character e in terms of multiplicative characters.Thus, if m=k M=K where M; K 1, then

e ÿmk

� � e ÿM

K

� � 1

fK X

x mod K

t ÅxxÿM 5:6

where, as usual for a character x mod K,

tx XKa1

xaea=K : 5:7

Note, for use in § 8, that tx0 mK , where x0 is the principal character modK, so that the contribution of the principal character to this expression foreÿm=k is just mK =fK . Now we wish to reduce the sum to a sum overprimitive characters. If w mod q induces x mod K, then (see [6, p. 67])

tx m Kq

� �w

K

q

� �tw: 5:8

Thus, by (5.7),

e ÿmk

� � 1

fK Xq jK

Xw

�m

K

q

� �Åw

K

q

� �t ÅwwÿM 5:9

since M; K 1 implies that M; q 1, whence xÿM wÿM; here andelsewhere the * indicates that the sum is over all primitive characters mod q. (Thecharacter mod 1 which induces all other principal characters will be included as aprimitive character.) Finally, we wish to eliminate the dependence in our formulaon g gcdm; k. By the MoÈbius inversion formula, for any f ,

f M; K f mg;k

g

� �Xd jg

Xe jd

md

e

� �f

m

e;k

e

� �: 5:10

The condition d jg is equivalent to d jm, d jk. Thus,

e ÿmk

� �Xd jmd jk

Xe jd

md=efk=e

Xq j k=e

Xw

�m

k

eq

� �Åw

k

eq

� �t Åww ÿm

e

� �

Xq jk

Xw

�t Åw

Xd jmd j k

Xe jd

e j k=q

md=efk=e

Åwÿkeq

� �w

m

e

� �m

k

eq

� �

Xq jk

Xw

�t Åw

Xd jmd j k

wm

d

� �dq; k; d;w; 5:11


where for a character w mod q we de®ne

dq; k; d;w Xe jd

e j k=q

md=efk=e

Åwÿkeq

� �w

d

e

� �m

k

eq

� �: 5:12

Note that if k is squarefree, then

jdq; k; d;wj<Xe jd

e jk=q

fefk

d; k=qfk : 5:13

Now by (5.4) and (5.11), if we replace k by kq and m by md and rearrange,then

Mn Xq < y

Xw

�t Åw

Xk < y=q

bkqkq

Xd j kq

dq; kq; d;wX

m < kqT=2pd

anmd wm

Xk < y

1

kNn

y

k;kT

2p; k

� �; 5:14

where

Nny; z; k Xq < y

bkqq

Xw

�t Åw

Xd j kq

dq; kq; d;wX

m < qz=d

anmd wm: 5:15

To estimate this we distinguish the cases q 1, 1 < q < h LA for someA > 0, and h < q < y < y. The main term arises from q 1; the case 1 < q < h istreated using Siegel's Theorem, and the case h < q < y depends on an identity forAn and the large sieve. This is analogous to the proofs of Bombieri's Theoremgiven by Gallagher [7] and Vaughan [17].

In § 6 we consider small values of q > 1; in § 7 we consider larger values of q,and in § 8 we evaluate the main terms.

6. Small values of q

By (5.2) and (5.3),

anmd p t4mt4d 1 log m3 n 1; 2: 6:1Therefore, by Perron's formula (Lemma 4) with U explog w1=2,T p w p T 2, and d p T we haveX

m < w

anmd wm 1

2pi

Z viUvÿiU

Ans;w; d wsds

s O t4d

w

ULC

� �; 6:2

where v 1 log wÿ1, C > 0 is some constant, and

Ans;w; d X1m1

anmd wmms

; j > 1: 6:3

We apply Lemma 3 to show that Ans;w; d has an analytic continuation to the


left of j 1. We have by (5.2),

A1s;w; d X

d1d2d

X1m1

Lmd1wmms

! X1m1

m;d11

wm log md2ms

0B@1CA

X

d1d2dId1

L0

Ls;w ÿ Ld1

1ÿ wPPÿs� �

d s2d

ds

Ls;wFs;w; d1d s2

� �;

6:4where

Id 1 if d 1;0 if d 6 1;

(

Fs;w; d Yp jd1ÿ w ppÿs

Xn jd

mnwnns

;

and, if Ld1 6 0, then P Pd1 is such that Ld1 log P. Similarly, by (5.3),

A2s;w; d X

d1d2d3d4dId1

L0

Ls;w ÿ Ld1

1ÿ wPPÿs� �

´ d s2d s3d

ds

Ls;wFs;w; d1d s2

� �´

d

ds

Ls;wFs;w; d1d2d s3

� �Bs;w; d1d2d3; d4; 6:5

where

Bs;w; d; e Xm;d 1m < y=e

bmewmms

: 6:6

Thus we see that Ans;w; d has a meromorphic continuation to the whole plane.If w is the principal character mod 1, then Ans;w; d has a pole at s 1. For anyother character, Ans;w; d has at most one (simple) pole in the region

j > jot : 1ÿc

log q1 j t j ;

where c is an absolute constant. By Siegel's theorem, this pole, if it exists, is at areal number b which satis®es

1ÿ b q« qÿ« 6:7for any « > 0.

We will move the path of integration in (6.2) to the segment j joU ,j t j< U and estimate the integral on the new path. In doing so, we cross the poleat s 1 if q 1 and the pole at s b, if it exists. De®ne

Rnw; d Ress1

Ans; 1; d ws

s: 6:8


Let G be the path consisting of three line segments fj ÿ iU: v > j > jog,fjo it: ÿU < t < Ug, and fj iU: jo < j < vg. Then by (6.2), (6.8), andCauchy's theorem,X

m < w

anmd wm ÿ IqRnw; d

p

ZG

Ans;w; d ws

sds

�� Ressb Ans;w; d wss�� t4d wU LC: 6:9

By (6.4), (6.5), (6.6), and standard estimates,

Ans;w; d p t4d LCy1ÿjo

for j > joU , j t j< U, jsÿ 1jqLÿ1, and jsÿ b jqLÿ1, where C > 0 is anabsolute constant. Moreover, by the de®nitions of U, w, and jo,

wjoUpA w expÿCAL1=2for q < h LA , where CA denotes a positive function of A not necessarily thesame at each occurrence. With the choice

« 1=2A;it follows from (6.7) that

wb pA w expÿCAL1=2for q < h. Hence, by (6.9),X

m < w

anmd wm ÿ IqRnw; d pA t4d w expÿCAL1=2 6:10

uniformly for T p w p T 2 and q < h LA, where A > 0 is any ®xed constant.We use this in (5.15). Now

jtwj q1=2 6:11and by (5.13), if kq is squarefree, then

jdq; kq; d;wj< d; kfkfq : 6:12

Further, for kQ squarefree, Q p T , k < y, and j 12

or j 1, we have1

fkXd jkQ

trd dÿjd; k 1

fkXd j k

trd d 1ÿjXe jQ

treeÿj

ptrkkÿ1L if j 1;trkkÿ1=2 expCL1=2 if j 12 ;

(6:13

by Lemma 10 and since k p fk log log k. Here r stands for a non-negativeinteger, which is bounded above and which is not necessarily the same at eachoccurrence. Thus, by (5.15) and (6.10)±(6.13), for w qz =d , z q T , and any


A > 0, we ®nd that

Nnh; z; k ÿ bkXd j k

d1; k; d; 1Rnz

d; d

� �pA z

trkkLC

Xq < h

q1=2 expÿCAL1=2

pA trkz

kexpÿCAL1=2: 6:14

We now consider the larger values of q.

7. Large sieve estimates

For larger q we use Perron's formula (Lemma 4) with U T 20, T p w p T 2,and v 1 log wÿ1 to obtainX

m < w

anmd wm 1

2pi

Z viUvÿiU

Ans;w; d wsds

s O«w«: 7:1

Then by (5.15), (6.11), and (6.12),

Nny; z; k ÿ Nnh; z; kp« maxQ < y

Xd j kQ

m2kQd; kfk

Z yh

xÿ1 dSx T «y1=2; 7:2

where

Sx Xq < x

q1=2

fqX

w

� Z viUvÿiU

Ans;w; d qz

d

� �s dss

�� 7:3for z p Ty. We use an identity for An which corresponds to Vaughan's identity.We have

An Hn In n 1; 2; 7:4where

Hn Hns;w; d An ÿ Fn1ÿ LG 7:5and

In Ins;w; d Fn ÿ Fn LG An LG; 7:6here, Fn is a partial sum of An,

Fn Fns;w; d Xm < u

anmd wmms

; 7:7

L Ls;w, and G is a partial sum of Ls;wÿ1,

G Gs;w Xm < v

mmwmms

: 7:8

For the parameters u and v we will take

u x2; v T 1=2: 7:9Now we haveZ viU

vÿiUAns;w; d

ws

sds

Z viUvÿiU

Hn Ins;w; d ws

sds: 7:10


By (6.4) and (6.5), Ls;wAns;w; d is regular for j > 0; hence the same is trueof In. Therefore, by Cauchy's theorem, if q > 1, thenZ viU

vÿiUIns;w; d

ws

sds

ZG

Ins;w; d ws

sds; 7:11

where G is the path consisting of three line segments fj ÿ iU: 12

< j < vg,f1

2 it: ÿU < t < Ug, and fj iU: 1

2< j < vg. We can easily bound the

integral along the horizontal segments of G. By (6.4) and (6.5),

Ans;w; d Ls;wpX

d1d2d3d4d1 jLs;wj jL 0s;wj3

´ jBs;wjYp jd1 pÿj2 log3 d; 7:12

where

Bs;w Xm < y

bmwmms

7:13

for some coef®cients b which depend on d1, d2, d3, and d4, but satisfy

bnp 1 7:14uniformly in n and the di. Thus, for some ®xed r,

An L p trd L31 jL j jL0j3jBj 7:15for d p T and j > 1

2. If 1 < q p T and j > 1

2, then, it is well-known that

Ls;wp q js j1=4; L0s;wp q js j1=4L 7:16and it is trivial that

Fns;w; d p« y1«; Gs;w; Bs;wp T 1=4: 7:17Since U T 20, it follows from these estimates and (7.11) thatZ viU

vÿiUIns;w; d

ws

sds

Z 1=2iU1=2ÿiU

Ins;w; d ws

sds OT ÿ1: 7:18

Now by (5.15), (6.12), (6.14), (7.2), (7.3), (7.4), (7.10), and (7.18),

Nny; z; k ÿ bkXd j k

d1; k; d; 1Rnz

d; d

� �pA;« trk

z

kexpÿCAL1=2 y1=2T «

z maxQ < y

Xd jkQ

gm2kQfkd

Z yh

xÿ1dHnx

z1=2 maxQ < y

Xd jkQ

gm2kQfkd 1=2

Z yh

xÿ1 dInx; 7:19


where g d; k,

Hnx Xq < x

q3=2

fqX

w

� Z UÿUjHnv it j

dt

v j t j ; 7:20

and

Inx Xq < x

q

fqX

w

� Z UÿUjIn12 itj

dt12 j t j : 7:21

To estimate Hn and In we can use HoÈlder's inequality and the large sieveinequality in the form

Xq < x

q

fqX

w

� Z UÿU

Xn

anwnnÿit��

��2

dt

a j t j

pX

n

n x2 log U janj2; 7:22

this holds uniformly for a > 12

(see Vaughan [17]).By (6.1), (6.3), (7.7), and (7.22),X

q < x

q

fqX

w

� Z UÿUjAnv it ÿ Fnv itj2

dt

v j t j

pXm > u

m x2Ljanmd j2mÿ2v

p 1 x2uÿ1LCt4d 2 7:23for some C. Similarly, by (7.8),X

q < x

q

fqX

w

� Z UÿUj1ÿ LGv it;wj2 dt

v j t j

pXm > 1

m x2LXe jme < v

mewew me

� ��2mÿ2v

p 1 x2vÿ1LC 7:24for some C > 0, since the sum over e is 0 if m < v and is ptm in any case.Hence, by Cauchy's inequality, (7.5), (7.9), (7.20), (7.23) and (7.24),

Hnxp trd LC1 x2vÿ11=2x1=2

p trd x1=21 xT ÿ1=4LC 7:25for some C > 0.

Now for an arbitrary f f s;w let

A f Xq < x

q

fqX

w

� Z UÿUj f 1

2 it;wj dt

12 j t j ; 7:26


so that Inx AIn. Then by (7.6), (7.15), and HoÈlder's inequality, we haveInxp AF 2n 1=2A11=2 AF 2n 1=2AL41=4AG41=4

t4d L5A1 jL j6 jL0j61=2AB41=4AG41=4: 7:27By (7.7), (7.8), (7.13), (7.22), and (6.1),

AF 2n p x2 uLCt24d ;AG4p x2 v2LC;

and

AB4p x2 y2LC;for some constant C > 0. Also, A1p x2L, and by the method of Montgomery[11] (see also Ramachandra [14]),

AjL jk; AjL 0jkp« x2T « k 2 or 4: 7:28We also need (7.28) with k 6 which is our sixth moment assumption referred to inthe introduction. This follows from GLH and this is the only place we use GLH.

Thus, by (7.9), (7.27), and the above, we ®nd that

Inxp« t4d T «x1=2x2 u1=2x2 v21=4 xx2 y21=4x2 v21=4p« t4d T «x2 x3=2T 1=4 x3=2y1=2 xT 1=4y1=2: 7:29

Next, by (7.25),Z yh

xÿ1 dHnxp hÿ1=2 y1=2T ÿ1=4trd LC: 7:30

By (7.29),Z yh

xÿ1 dInxp« y y1=2T 1=4 y1=2y1=2 T 1=4y1=2t4d T «: 7:31

Hence, by (7.19), (7.30), and (7.31),

Nny; z; k ÿ bkXd j k

d1; k; d; 1Rnz

d; d

� �pA;« trk

z

k

� �expÿCAL1=2 trk

z

k

� �hÿ1=2 y1=2T ÿ1=4LC

trkz

k

� �1=2y y1=2T 1=4 y1=2y1=2 T 1=4y1=2T «:We use this estimate in (5.14); after substituting y y=k, z Tk=2p andsumming over k , we have

Mn ÿ Rn pA;« T expÿCAL1=2 Thÿ1=2LC T 3=4y1=2 T 1=2yT «;for some ®xed C > 0 and h LA for A > 0 to be chosen. We take A 2A0 C and have for any A0 > 0,

Mn Rn OA0;«TLÿA0 T 3=4y1=2 T 1=2yT «:

This gives Mn Rn oT provided y p T v with v < 12 .The rest of the paper is devoted to estimating the main term.


8. The main terms

In the sequel all equalities should be taken as asymptotic equalities. We alsoassume throughout that y T v with v < 1

2. Since the contributions from q > 1 have

been shown to be negligible, only the principal character term, q 1, contributes.This essentially means that for the purposes of obtaining main terms one mayreplace eÿm=k by mK =fK (compare (5.6), (5.7), and the remark below(5.7)). Hence we may write the expressions for Mn more transparently as

Mn Xk < y

Xm < kT =2p

anmbkk

mK fK :

We now describe a simple way of obtaining the main term for M1 . We mayeasily recast our expression for M1 as

M1 Xl < y

Xk < y=l

bklkl

mkfk

Xm < kT=2pm; k1

a1ml: 8:1

Observe that a1n P

d jn Ld logn=d log2 nÿP

d jn Ld log d. We willreplace

Pd jn Ld log d by

Pp jn log

2 p; the overall error in evaluating M1caused by this is easily seen to be OT log T . Thus,X

m < kT =2pm; k1

a1ml X

m < kT =2pm; k1

log2ml ÿXp j l

log2 pÿXp jm p; l 1

log2 p0@ 1A

fk T2p

log klT 2 ÿXp j l

log2 p� �

ÿX

p < kT =2p p; kl 1

log2 pX

m < kT =2ppm;k1

1

fk T2p

log klT 2 ÿXp j l

log2 pÿ 12log kT 2

� � fk T

2p12log kT 2 2 log l log kT log2 lÿ

Xp j l

log2 p� �

:

Employing this in (8.1) we obtain M1 M11 M12 M13 M14, with obviousmeaning. Using Lemmas 5 and 6 and integration by parts we see that

M11 T

2p

Xk < y

mkk

log kT 22

Xl < y=k

bkll

T2p

Xk < y

m2kfk

log kT 22 log y

P 0logy=k

log y

� �


T2p

log2 y

2

Z 10

P 01ÿ a1=v a2 da

T2p

log2 y

2vÿ2

Z 102=v 2aP1ÿ a da

� �:

Similarly,

M12 2T

2p

Xk < y

mkklog kT

Xl < y=k

bkll

log l

ÿ 2T2p

Xk < y

m2kfk log kT P

logy=klog y

� �

ÿ 2T2p

log2 y

Z 101=v aP1ÿ a da:

It is trivial from Lemma 5 that M13 0. Finally,

M14 ÿT

2p

Xk < y

mkk

Xp < y=k

log2 p

p

Xl < y=kp

bklpl

ÿ T2p

Xk < y

m2kfk

Xp < y=k

m p log2 p pÿ 1 log y P

0 logy=kplog y

� �

T2p

log2 y

Z 10

Z 1ÿa0

bP 01ÿ aÿ b db da

T2p

log2 y

Z 10

Z 1ÿa0

P1ÿ aÿ b db da

T2p

log2 y

Z 10

aP1ÿ a da:

Adding the expressions for M11 , M12, M13 and M14 above, we obtain

M1 ,T

2p12L2 ÿ

Z 10

Px dxL log y� �

;

so that by (3.13), and (5.5) we have the required estimate

S1 T

2p12L2 L log y

Z 10

Px dx� �

O«T 1=2«y oTL2:

We now turn to the task of evaluating M2. As in (8.1) we may recast ourexpression for M2 as

M2 Xl < y

Xk < y=l

bklkl

mkfk

Xm < kT =2pm; k1

a2ml

Xl < y

Xk < y=l

bklkl

mkfk SkT =2p; l; k; 8:2


say. Let Ljn be de®ned byP

n Ljnnÿs ÿz 0s=zs j and gn byXn

gnnÿs ÿz 0s=zsz 0s2 ÿz 0s=zs3z2s:

Lemma A. Suppose a and b are coprime, squarefree integers and that0 < j < 3 is an integer. Let gjn; a; b be de®ned by the relationX1

n1gjn; a; bnÿs

Xn;ab1

Lnnÿs� �j X

m;b1dmamÿs

� �:

Then

Gjx; a; b :Xn < x

gj n; a; b, xlog x j1 j 1!

fb2b2

da;

where da Pp ja2ÿ 1=p. Further,

Gx; a; b :Xn < xn;b1

gan, x fb2

b2

X3j0

3

j

� �bjada

log x j1 j 1! ;

where bja :P

d ja L3ÿjd =dd .

Proof. By standard ideas, the main term is the residue at s 1 ofxs

sÿ z

0

zs

Xp jab

log p

ps ÿ 1

!jz2s

Yp jb1ÿ pÿs2

Yp ja1ÿ pÿs22 3pÿs . . .;

which is asymptotically equal to the right-hand side of the ®rst statement. Thesecond assertion follows from the ®rst upon noting thatX

n;b1Lnnÿs

� �3X3j0

3

j

� � Xn;ab1

Lnnÿs� �

jXn;a> 1

Lnnÿs� �3ÿj

:

From the de®nition of a2n we easily see thatSkT =2p; l; k ÿ

Xm < ym; k1

bmGkTl; m=2pm; l=l; m; k

ÿXl1 j l

Xm < yl1=lm; kl 1

bml=l1GkT =2pm; l1; k:

By Lemma A and Lemma 5, the right-hand side above is

,ÿ Tf2k

2pk

X3j0

3

j

� �Xl1 j l

dl1bjl1X

m < yl1=lm;kl 1

bml=l1m

logkT =m j1 j 1!

,ÿ Tf2k

2pk

X3j0

3

j

� �1

j 1!Xl1 j l

dl1bjl1ml=l1kl

fkl


´log kT j1

log yP 0

logyl1=llog y

� � j 1log kT jP logyl1=l

log y

� ��

,ÿ fklT2pfl

X3j0

3

j

� �1

j 1!Xl1 j l

dl1bjl1ml=l1log kT j

´log kT

log yP 0

logyl1=llog y

� � j 1P logyl1=l

log y

� �� :

Using this in (8.2), making obvious substitutions, and interchanging summa-tions, we see that

M2 ,ÿT

2p

X3j0

3

j

� �1

j 1! M2 j; 8:3

where

M2 j :Xl1 < y

bjl1dl1fl1

Xk < y=l1

mkklog kT j

Xl < y=kl1

mlfl blkl1

´log kT

log yP 0

logy=llog y

� � j 1P logy=l

log y

� �� : 8:4

A straightforward argument shows that the innermost sum over l is equal to

mkl1X

l < y=kl1l; kl11

m2lfl P

log y=kll1log y

� �log kT

log yP 0

log y=l

log y

� � j 1P log y=l

log y

� ��

, mkl1fkl1

kl1log y

log kT

log yQ0

log y=kl1log y

� � j 1Q1

log y=kl1log y

� �� ;

where for i 0, 1, Qia :R a

0 Paÿ bP1ÿi 1ÿ b db. Employing this in (8.4)we obtain

M2 j, log yXl1 < y

dl1bjl1ml1

l1

Xk < y=l1k; l11

m2kk2

fklog kT j

´log kT

log yQ0

log y=kl1log y

� � j 1Q1

log y=kl1log y

� �� : 8:5

A simple calculation reveals that the sum over k above is

, log y j1Rjlog y=l1log y

� �Qp1 pÿ 1=p21ÿ pÿ1Q

p j l11 pÿ 1=p2

log y j1Rjlog y=l1log y

� �C

Cl1;

where C and Cl1 have their natural meaning andRja

Z a01=v b j1=v bQ0aÿ b j 1Q1aÿ b db:


Substituting this in (8.5) and recalling the de®nition of bjl1, we have

M2 j, Clog y j2Xl1 < y

dl1bjl1ml1

l1Cl1Rj

log y=l1log y

� �

Clog y j2Xd < y

md L3ÿjd dCd

Xl1 < y=dl1;d 1

ml1dl1l1Cl1

Rjlogy=l1d

log y

� �: 8:6

Mimicking the argument of Lemma 10 of Conrey [2] we see thatXl1 < y=dl1;d 1

ml1dl1l1Cl1

Rjlogy=l1d

log y

� �, Cÿ1

Yp jd

p2 pÿ 1p2 ÿ p

1

log2 yR 00j

logy=d log y

� �:

(It is an easily veri®ed crucial point in this argument that Rj0 R 0j0 0.)Plugging the above display into the right-hand side of (8.6), we obtain easily

M2 j, log y jXd < y

md L3ÿjd d

R 00jlogy=d

log y

� �:

Using the above display and the familiar distribution of the L3ÿjn (andintegration by parts) we arrive at

2pM2

Tlog y3 ÿX3j0

3

j

� �M2 j j 1!

1

log y3

Z 1

0

12a2R 0001ÿ a daÿ

3

2

Z 10

aR 0011ÿ a da

36

Z 10

R 0021ÿ a daÿ 124R 0031

Z 1

0R0a daÿ 32R11 12R 021 ÿ 124R 0031:

An easy computation shows that the expression in the right-hand side abovemultiplied by v3 equals

ÿ 124v

Z 10

P 0x2 dx 112 3v

2

Z 10

Px2 dxÿ v2

Z 10

Px dxÿ v2

2

Z 10

Px dx� �2

:

This completes our treatment of M2.The estimate (2.7) now follows from (3.21), (3.26), (3.27) and the above, and

so proves our theorem.

References

1. A. Y. Cheer and D. A. Goldston, `Simple zeros of the Riemann zeta-function', Proc. Amer.Math. Soc. 118 (1993) 365±373.

2. J. B. Conrey, `Zeros of derivatives of Riemann's y-function on the critical line', J. NumberTheory 16 (1983) 49±74.

3. J. B. Conrey, `On the distribution of the zeros of the Riemann zeta-function', Topics in analyticnumber theory (University of Texas Press, Austin, 1985), pp. 28±41.


4. J. B. Conrey, `More than two-®fths of the zeros of the Riemann zeta-function are on the criticalline', J. Reine Angew. Math. 399 (1989) 1±26.

5. J. B. Conrey, A. Ghosh and S. M. Gonek, `Mean values of the Riemann zeta-function withapplication to the distribution of zeros', Number theory, trace formulas and discrete groups(Academic Press, Boston, 1989), pp. 185±199.

6. H. Davenport, Multiplicative number theory, 2nd edn (revised by H. L. Montgomery), GraduateTexts in Mathematics 74 (Springer, New York, 1980).

7. P. X. Gallagher, `Bombieri's mean value theorem', Mathematika 15 (1968) 1±6.8. S. M. Gonek, `Mean values of the Riemann zeta-function and its derivatives', Invent. Math. 75

(1984) 123±141.9. D. R. Heath-Brown, `Simple zeros of the Riemann zeta-function on the critical line', Bull.

London Math. Soc. 11 (1979) 17±18.10. N. Levinson, `More than one-third of zeros of Riemann's zeta-function are on j 1

2', Adv. in

Math. 13 (1974) 383±436.11. H. L. Montgomery, Topics in multiplicative number theory, Lecture Notes in Mathematics 227

(Springer, Berlin, 1971).12. H. L. Montgomery, `The pair correlation of zeros of the zeta function', Analytic number theory

(St Louis University, St Louis, Mo., 1972), Proceedings of Symposia in Pure Mathematics 24(American Mathematical Society, Providence, R.I., 1973), pp. 181±193.

13. H. L. Montgomery, `Distribution of the zeros of the zeta function', Proceedings of theInternational Congress of Mathematicians (Vancouver, B.C., 1974), vol. 1 (CanadianMathematics Congress, Montreal, Quebec, 1975), pp. 379±381.

14. K. Ramachandra, `A simple proof of the mean fourth power estimate for z12 it and

L12 it;x', Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1 (1974) 81±97.

15. P. Shiu, `A Brun±Titchmarsh Theorem for multiplicative functions', J. Reine Angew. Math. 313(1980) 161±170.

16. E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd edn (edited and with a prefaceby D. R. Heath-Brown, The Clarendon Press, Oxford University Press, New York, 1986).

17. R. C. Vaughan, `Mean value theorems in prime number theory', J. London Math. Soc. (2) 10(1975) 153±162.

J. B. Conrey and A. Ghosh S. M. GonekDepartment of Mathematics Department of MathematicsOklahoma State University University of RochesterCollege of Arts & Sciences Rochester401 Mathematical Sciences NY 14627Stillwater U.S.A.OK 74078-0613U.S.A.

E-mail: [email protected] [email protected]@math.okstate.edu

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