Analytic Number Theory: Proceedings of a Conference in Honor of Heini Halberstam [2] 0817638245, 0817639330, 0817639322, 3764338245, 3764339330, 3764339322

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Progress in Mathematics

Analytic | Number Theory Proceedings of a Conference in Honor of Heini Halberstam Volume 2 = Bruce C. Berndt Harold G. Diamond Adolf J. Hildebrand Editors

Cs (at :

aS

Birkhauser

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FLORIDA STATE UNIVERSITY LIBRARIES

JUL 15 1997 TALLAHASSEE, FLORIDA

Progress in Mathematics Volume 139

Series Editors

Hyman Bass Joseph Oesterlé Alan Weinstein

Analytic Number Theory Volume 2

Proceedings of a Conference In Honor of Heini Halberstam

Bruce C. Berndt Harold G. Diamond Adolf J. Hildebrand Editors

Birkhduser Boston ¢ Basel ¢ Berlin

Bruce C. Berndt

Harold G. Diamond Adolf J. Hildebrand Department of Mathematics University of Illinois Urbana, IL 61801

Se} Q f pee CYT | A

7

4B6

[99 CG Vv. ae )

Printed on acid-free paper Se

© 1996 Birkhauser Boston

Dirkhiduser

®

ip

Copyright is not claimed for works of U.S. Government employees. Allrights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhauser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. ISBN 0-8176-3824-5 Vol. 1 ISBN 0-8176-3933-0 Vol.2 ISBN 3-7643-3824-5 Vol. 1 ISBN 3-7643-3933-0 Vol.2 Typesetting by the editors in TEX Printed and bound by Quinn-Woodbine, Woodbine, NJ Printed in the U.S.A.

Shes (. s)4) 3) 7

ISBN 0-8176-3932-2 SET ISBN 3-7643-3932-2 SET

Contents

of Volume

Pontents Ot Volumes) trian Goes eet taeents PCAC

ee

cert

erin

et

ed cic PO Ce

LASh Ol; PAT AGIAN.

4

ck

Me Um Se

NEY COIN WEGNER Ride gedl cyt Acai chia

ato

2

CR

VO

ee,

CP OGY

occ tas

oto

hea

dak peo

bas ix

Teer hee carne renee

a Rae

og

I

Pe

xi xiii

451

Cas cen evens RoeniCks 465

PEL a te cee ee Ook Matis si. sinensis in oes hice eecw ate ik eee On an elementary inequality in the theory of Diophantine approximation

Ju DOES sls Bird Fs RON ae eg ergael ACR keg ee The integer points close to a curve II Me JUtilas..

BD, vii

Mee oi

Alte ARE Ce Cacti wissen e Dances

|W RFS 8 Coe Sl Se gee ae Cee Ck a a An estimate for Heilbronn’s exponential sum Wes Ere liam ere a Pies ere PIPES On B3 sequences

POR SE

eee nce ee

eee

471

neta 487

s eee er, ee ea Te LOM IR eee sis. orn Taek The fourth moment of Riemann’s zeta-function and the additive divisor problem

Ife, Ju IeUNEPARN AVG As ANEIVE? ot.3c:cn0.0 Bag b OOo Oe oTeee Oe ae On the uniform distribution of Gauss sums and Jacobi sums

yes 517

eo our 537

WG, INGE LRG) KOnA eee nen aon Ae eS an Oe OA ae eee Manne ee eR mere ens 559 A problem of Steinhaus: Can all placements of a planar set contain exactly one lattice point?

ee

RAE PARE? SALAAM OSes SSG CN es ie A note on the Riemann-Roch Theorem for function fields

SEs

567

Jos LencugenoVe ea EVAN Ree on oerae aoe eae Oeiac cic ke cn ea RE Or ere Catto, cca Sil Estimation of exponential sums over primes in short in-

tervals IT

Le? Liteht ameeniw reCueicatintn. estos ae eee, Weighted Wiener-Lévy theorems

PE) ae

ERE

Se es 607

NY VERTPoy 8 winborers A Gud Ginbag oir REPT COAG OD.OO LE ata Rae CETORem e oe te Spectral mean-values of automorphic L-functions at spe-

621

cial points jis USPS Ris 8 Bele. ie Soi ST aR

eeDR EO re RN eee Cre The size of the coefficients of cyclotomic polynomials

a

633

Contents

vl

Volume 2

econo Uadnut ToueoDdGomo asd oor mine GaN V Toire)akekaliners ears cin Oo On Kuznetsov’s trace formulae

641

an err 669 M. Re Murtynd CoS. Rajan .....00. oe, eens oes 8 ae Stronger multiplicity one theorems for forms of general type on GL2 Me Naircand AS Perelli.) ac cecrsarse it eaitatietrior shelter eleieie kien taee Rees 685 A sieve fundamental lemma for polynomials in two vari-

ables Ce Pomerance sas oo Ss easatpeneeevorsctask vn see ee ade tea Multiplicative independence for random integers Ri As Rankines aciig cvoaiscn orotate, oe si oreseaeay sees oe PERER On certain meromorphic modular forms

e

osioe ons703

eceeatemt ys kotor (Als)

Bs Scour fel deck a080 te Soros grratteocud sone ei eens pot oe enor iee ice Rae te chee 723 Comparison of two dissimilar sums involving the largest prime factor of an integer

HY Mie Stark occ. eas cs.sis,x-cls. obo 3 SB aneertar de Se:

Aeron aR

eee

eee con

On the determination of an L-function from one value Ky Bi Stolarskyaean ace gai Shoes we atone ee aeeae IO ee oy ise An approximation to the q-analogue ofn involving the n-analogue of a golden number

Re Gy Vallig Rani nee eta timtantan aay SA. clk es eA Small values of Dirichlet L-functions at 1

aE

oe 745

cess csS

eet ean eee Saeco ONS Global zeta functions over number fields and function fields

755

BBVA, isidscecest cede Seuesint oars

767

RES VNONDS Gace atckc) date pela ora Sosa ste Sak rere eT Consequences from the study of concentration functions on shifted twin primes

CE

SB.

WOOLEY ar creearil cases ofits atcdayn sve «:otsgstoiveas Ser ete ee An affine slicing approach to certain paucity problems

803

DO,

BaD

817

he ys eas ea RRL eee a Ramanujan’s class invariants, Kronecker’s limit formula

and modular equations (II) WB

Zhang $i

0d oa

ees oe ee, Probabilistic number theory in additive arithmetic semigroups I

839

Contents

Pee AIS

of Volume

1

eee ey pO CEs ah Sm Rene rae mma ee Weighted partition identities and applications

ae

care.

aed Dye TITERS5 Re ER gn any, hla UN” a rt ee mR Al ate tarae ee Rogers-Ramanujan polynomials for modulus 6

Bee

Hes

ee tee Soe ME Be eee NT SO On subset-sum-distinct sequences

SoM

oe Lee

ile

care eee 31

dket ANG ge Haraairectarrrcs aire con tener: oo sae oleae 39 The Brun-Titchmarsh Theorem on average

oe

MLS DATION SUG wie WOTay aces oer ie eee one otis Reamer trator 105 Congruences for Fourier coefficients of half-integral weight modular forms and special values of L-functions

teed 8g Sy NTS Teh ek Dt 8 OP es Be I Sia i fn a as a The asymptotic formula for the number of representations of an integer as a sum of five squares

129

UE BTS vey AN Yo A Sag To)Vg 0 eens een Meee eee et eee ee eee Distribution of the error term for the number of lattice points inside a shifted ball

eee 141

INPREIS OSG COU epee pete chee rt ete aeseeeee cee oye a eg, gucrai'e re testederesevsncunveleteeuetnvaeioeon OS 155 A probabilistic generalization of the Riemann zeta function JD). BOMe INNIS SidSome eae

Rome Oe eias Cee ee cre Mee Seema e mame 163 A general Heine transformation and symmetric polynomials of Rogers

DD SISTAGleye ck ane ceminmace pels Nears A sieve auxiliary function

ate As cone

ote

cs

DAC OCHIARE Ais cashesorsSes PAL ects On eae este oe oe be See Bounds on complete exponential sums

oe

as wath ee

Ue So Cotibes oetee itenetam ia ada. tiem eer axial 20-0 5 Shrne le wees A note on the fourth power moment of the Riemann zeta-function

ad 173

Lae

kN

211

Die225

Ete DAD Gussie eee eto nico tae cieic ase peestreh bc GG Re Manuine ee nen eree 231 Effective estimates of exponential sums over primes

(BL BPS EVa(re eeiest ety teh nee a nar earn eee = or ee ee ee ee Se eee 245 On products of multiplicative functions of absolute value at most 1 which are composed with linear functions

viii

Contents

Volume

H: G; Diamond; H: Halbétstam, and HB tienert.. 5. i..k csv een tes Combinatorial sieves of dimension exceeding one II 1D: Bichhornrangt he One -sere tris eek ae ieee Congruences for partition functions

P.DVTCA. EIGtt. scoetacashengn bate ooa bed Fractional power large sieves Po EES

acs oieenecanes Bros gpeve Gt

eve

iN

ae

eette

OR

eee

Man

265

rate tote 309

Debits. 323

Ot ELUTE POG EM as Olek 333

Some problems I presented or planned to present in my short talk . P. Erdos, S: W. Graham, As ivic-and’C. Pomerance suret¢ On the number of divisors of n! RES VaMns 5.285 Sok Recess ts corecetotayaves eee eye ere Generalized Lambert series M, Pilaseta;qgsas:

LR

ene atedsa

mcs cao re

eT

337 357

yh dvtend aie una toute Gay Aare RR Selia-a.ts 371 A generalization of an irreducibility theorem of I. Schur

Mie Tao Hlanive Boch). WOOUS mn cot. je secre aiindata ae eek Small values of indefinite binary quadratic forms See CICOLATAb Rene EL aLWATHOG DORON ee NOR he ooh

SH EME UP RV ZH OO SEH SEH QEAPPOUP PNOMENEQOORY

List of Participants

a

Kovacs (Univ. of Illinois) . Laporta (Univ. of Toronto) Li (Purdue Univ.) Lou (Halifax) . Luo (MSRI) Martin (Univ. of Michigan) Meyer (Univ. of Illinois) Morée (Macquarie Univ.) Motohashi (Nihon Univ.) Murty (McGill Univ.) . Nathanson (Lehman College) M. Nicolas (Université Lyon I) Peral (Univ. del Pais Vasco) . Philipp (Univ. of Illinois) Pomerance (Univ. of Georgia)

Prather (Lexington, KY) Ream (Univ. of Colorado) Reti (Univ. of Florida) Robins (Univ. of North. Colorado) Scourfield (Royal Holloway Coll.) Shiu (Loughborouh Univ.) Skinner (Princeton Univ.) CS Be eee po, jm Se etic eee aot ee ae Uae ay Srinivasan (Univ. of Georgia) =— Stark (MSRI) Stolarsky (Univ. of Illinois) Thunder (Northern Ill. Univ.) C. Vaughan (Imperial College) Walling (Univ. of Colorado) Wan (Univ. of Nevada) . Wijsmuller (La Salle Univ.) Wolke (Univ. Freiburg) Wooley (Univ. of Michigan) C. Zhang (Southw. Miss. St. Univ.) PTAA AEN eae Zheng (Zhongshan Univ.)

Kwok (Univ. of Illinois) (Univ. of Illinois) Li (Univ. of Georgia) Lucht (TU Clausthal) Maier (Univ. of Ulm) Methfessel (Univ. of Waterloo)

1, we note that the last two terms in these are essentially the same, while the first and second terms in the former are, respectively, better and worse than the second and first terms in the latter. Thus there are occasions when Theorem 1 is superior to Lemma 1. In the third application of this alternative technique to our problem it is necessary to use a refined version of Theorem 1 that depends on a more precise form of (11). 5. Approximating rationals having kth power denominators — preliminary analysis of the problem The genesis of the investigation is the introduction of two sums of the type Nw(¢,y) that count the number of positive integers m not exceeding y for which there is an integer @ satisfying the conditions

(12)

nr Al Ready,

(Cran) = 1,

where it will be assumed throughout that k > 2. First, for each of the infinitely many rational approximations p/q in lowest terms to the given irrational number @ for which

(13)

| — p/q| < 1/4?

and q is sufficiently large, we choose a large number x and a number v = v(z) such that

(14)

Qua" =q"

and

(15)

F — ea)

or) |lah > =e Aer y a

where the right-hand sides of the inequalities are not less than 1 by (21), we must allow for the additional term 1(k-1),.4(k—2) [2 r4 O | ——__—_——_ (

va

i:

which will induce an extra contribution of

,) (“—k a /

)

/

a

1/K’'

oe

ya

yl/2K

to M,—1(2u, 2v;m*p/q) and thus one of

pit(k-2)/2K+6

(31) to a

O (Soa) re

In the opposite case where both inequalities in (30) are negated, we analyze separately the situations in which ks y2

(32)

tk Tie

aocis L2il or |n| $< —,

initially considering in the first case the influence of any 7 for which

(33)

&

nie 20..

Then, for such a value of 7, any extra term of type (28) is 1k

G2

pv20

which by way of (8) leads to a donation of

pK rR “1p bee

(34)

ON.

a | eee ea V20

ie

i

rae

pi-2k/K gh/K+

=a

\eeominke a7

484

C. HOOLEY

to My_1(2, 20; m*p/q).

But, if (29) hold when m < a/p < q, then (u,q)

and (w,q) have a common value e dividing g. Hence, setting u = eu’, w = ew’, where (u’,q/e) = (w’,q/e) = 1, we have pm*u' = w', mod q/e, through which congruence a given pair u, w determines m*, mod q/e, and hence corresponds to at most O (K(a/e)) =U (K(9)) incongruent values ofm, mod q/e and thus to at most O (ek#(9)) values of m not exceeding x/y. Thus, since the number of appropriate m that answer to all pairs in (29) such that 0 < u < p; and 0 < |w| < p2 is therefore

O

ke)

Se

elg.

s

|= @

RO

pp»

0 Gant

In this case our construction reduces to that of Bambah and Chowla. The general statements of the theorems follow. Theorem 1 deals with the case n = 2, and Theorem 2 with the cases n > 3. In Theorem 3 we take n = 2, and restrict m to a subsequence of the integers in J. In Theorem 4 we construct an integer point close to a space curve satisfying certain conditions.

We write [t] for the integer part of a real number ¢, ||¢|| for its distance from the nearest integer, and (a,b) for the highest common factor of two integers a and b. Theorem 1. Let M > 12 be a positive integer, and let I be a closed interval of length M with integer endpoints. Let f(x) be a real function, twice continuously

THE INTEGER

POINTS CLOSE TO A CURVE II

489

differentiable on I, with

(1.1)

AAS wi ia sl’ @lscA 4 and

(1.13)

6

(1.14)

HOC

sos 18wo HC” M

(1.15)

2+1/L

Cy ( ) 9/2 C?

928/232T,

Then there are distinct integers mj,...,MR

(1.16)

in I with

If (mr) < 6,

and fori =1,...,n—1

there are integers k;, and powers of two qj, such that

(1.17)

OC

k;

0)

eee | Vir

K

and for’ = 2,...,n—1 (i)

(1.18)

bi

(mr) =

Kir

ih

< 36L(C

Ba

A)

Vier

’

with

KES

(1.19)

Vir S HLC3A:

Here

‘

gy Or

(1.21)

6°>M

bags Ga

dA

aT)

=a.

po ae|

L

Oe 93/232 HL O2 95/232

:

C2

bai

ide SN ee 925/232 KT,

THE INTEGER POINTS

CLOSE TO A CURVE II

491

forn > 4. Theorem 3. Let M > 12 be a positive integer, and let I be a closed interval of length M with integer endpoints. Let f(x) be a real function, twice continuously differentiable on I, with

v22 (1.22)

|

i

1 — SB A SMS 1 and A i +1

(Kn)! Deen

k+1 (3.17)

1/2

for i > 3,

cn

a

rave i(k)

i!

k+1

4

4/2*

< .2'32(CgA)!/?" Phe a!

3

< 9k 32 C(2k+2)/2" (CqA)*/2" e4

< 2k+63202(CgA)!/2 < 1/64K by (3.15), (3.9) and (3.2). From (3.16) and (3.17) we see that 1

(3.18)

1

g(Kn) — ko — 5 hak en® aS rae

Wie

Let J be the interval of values taken by g(x) + a0/K for0 2 we use 2¢ > 2i-to get

(K,N)'A; < 2*3? ( aA (ota)

)

by (3.9) and (3.39). By (3.30) the factor raised to the i-th power is less than 1/2, 80

GON dX a k

(ookehp a

z

asi nae Bom Eeey,

ees

ae

at A)V/

TL seta aia

pas

RA)

I

Hence we have

(3.42)

| me

Sa

< ok+293

SAE |32K3R gree3 (CM

k ee

6 es big

and for |n — n’| < qK» we have

(3.43)

IG(n) — G(n')| < 6.

We are now ready for the choice of n. By (3.38) and (3.40)

(3.44)

G(N')

> -——__ - —-—

+ 57

> ro.

Let n, be the smallest integer with

G(n,) > T2, Nr = bs (mod qgk2). Then n, < N by (3.38) and (3.44), and n, > 0 by (3.37). Ifn, < qKo, then

G(0) < re < G(n,) < G(0) + 6. If n, > qKo, then

G(np — qK2) < rz < G(nr) < G(np — qK2) + 4, a in both cases |G(nr) 7 r9| < 6,

M. N. HUXLEY

512

which verifies (3.33). We must also check that different integers r and r’ cannot give the same value of n. If np = np’, then r, = 71, and

Lass = r2| “< 20.

Since (3.30) gives 6 < 1/2, we see that r2 = rj, so the integer n, determines r uniquely, and our R points are distinct. For the derivatives we use ~

’ fO(m + Kin) = fO(m) + Kynf 9+) (m) +--+

k-j ey fi _— 4]

(Kyn)h

(k -—j+1)!

)

!

(k+1) (py)

ees ea q

’

for some 7 depending on n and J, with

ee

K

Cp =O;

a rational number.

:

a

(Kin)*~9

hea

Since kK, is the odd part of k!, the denominator of ¢; is at

most the power of two dividing (k — j)!, ((K — 7)!)2 in the notation of Lemma 6. We have

Pj

=;

+

Kinkjes

oe

+

(Kan)ny

le aM

(Kin)

gf?)

govt

so that for 7 >1

|S Age

lp;

gi

(Kyn)*-3+1

1

NA ge pe j

(k—-j+)!

kes

For j = 1 this is the sum estimated in (3.40) and (3.41), so

p1| < 6/K,

which verifies (3.34). For j > 2 we use (3.7), (3.9) and (3.39) with 2*>¢+1

THE INTEGER POINTS

CLOSE TO A CURVE II

513

for t > 0 to get

tone’ Dal gre aly lca k—j+1

t/2

b/2k

1

t

—jt+l1

One

x}7" for some T;, T2 = T. The resulting integral, say I(T,,T2), then takes two different forms.

The left hand side of (1.5) yields T2 (2.2)

I(T,,T>2)

=

|

Ti

I(r) dr,

FOURTH MOMENT OF RIEMANN’S ZETA-FUNCTION

521

(2.3) I(r) = 271e267 ‘i

_

2 cosh mt

KG + it)|*e~ 7% ;

i

Ay §

Mares ere ere, bars

It is easily seen, as will be shown next in §3, that this is essentially the fourth

moment of the zeta-function over the interval [T,, 7]. On the other hand, the right hand side of (1.5) leads to the alternative formula

(2.4)

Td, 15) —

T af oe maelhe p(2rixe™— 5) b(—Qmize

Ut) e27ui—w? /U? dx du dr.

Ti

In 84, we shall see that this leads to the additive divisor problem, and in §6, a

spectral theoretic expression is given for the same integral. In subsequent calculations, repeated use will be made of the familiar formula

(2.5)

‘| eAt—Be* de = Nee =

(Re B > 0).

3. An analytic interpretation of the integral /(T,,7T>) To begin with, we simplify the equation (2.3) for I(r). Clearly, the integral over t can be restricted to [0,co) with an error O(U). Moreover, the factor e™'/2 cosh rt can be simplified to 1 and the integral over u can be extended to the whole real line. Then, evaluating the latter integral by (2.5), we have

(3.1)

1,72)

= [wl NICS + it)|* 0

dt + O(TU),

where T2

w(t) = aang

| exp(26(7 — t) — (r —#)°U") dr T) T2-t

=n ?y

Tint

exp(26u — v?U”) dv.

This shows that w(t) decays rapidly as the distance of t from the interval [T,, Tz] increases, and w(t) < exp(— log’ T) if t¢ [T; — A, T. + A]. Therefore (3.1) remains valid if the integral is truncated to the last mentioned interval:

(3.2)

1(0,Te) =f

Tot+tA

w(A)IC(5 + it)|4 dt + O(TU).

522

MATTI JUTILA If we extend the v-integral to the whole real line, then the resulting upper

bound for w(t) equals exp((6/U)?) = 1+ O((TU)~?) by (2.5). Also, this gives a good approximation to w(t) if ¢ lies in the interval [7), 7] not too close to its endpoints; more precisely,

(3.3)

w(t)

=1+O((TU)~*) for t € [T,; + A, Te — Al.

The fourth moment of the zeta-function over the interval [T;, 72] may now be compared with J(T,,72) as follows. First, by (3.2), (3.3), and the wellknown upper bound < T log* T for the fourth moment, we have T2-A

I(T,,T») + O(TU) > |

w(i)le(5 +at)|*at

TM14+A T2-A

= i

1

ISS tht Of

Olle Ue lO 1

™Mm4+A

Applied with 7; — A and T> + A in place of T, and T», this gives T: ye 631 (3.4) | IS + it)|* dt < I(T, —A,T. + A) +O(TU) + O(T—'U7 log* 7).

Ti

To derive an inequality in the reverse direction, note that by (3.2) and the properties of the weight function we have Tot+A

TT 205 es if

1 ISS + it)|* dt + O(TU) + O(T~1U~? log* T).

T,-A

Applied with T; + A and T2 — A in place of T, and T>, this gives T2

1

(3.5) a SGe: it)|* dt > I(T, + A, Tz— A) + O(TU) + O(T1U~ log* T). Finally, combining (3.4) and (3.5), we obtain

(3.6)

I(T, + A,T, — A) + O(TU) + O(T-1U~? log* T)

T:Z

1

.

= ii Seas it)|* dt < I(T, — A,T) + A) + O(TU) + O(T-1U-? log? T).

FOURTH MOMENT

OF RIEMANN’S ZETA-FUNCTION

523

4. An arithmetic interpretation of the integral /(T,,7>) Coming now to the core of our argument, we analyze the integral J(T,,T>) in its shape (2.4). For this purpose, an approximate functional equation for the

function $(z) (a slight generalization of eq. (7.16.2) in [T2]) will be needed.

Lemma 1. Let z lie in the domain a < |arg z| < 1/2, |z| > b for some positive constants a and b. Then for alle > 0 we have

(4.1)

o(1/z) = — sgn(Im z)2nizo(4n7z) + O(|z|*),

where the implied constant depends only on a, b, and e.

Our goal in this section is to prove the following arithmetic formula for

our integral. This is, essentially, Lemma 3 in [H-B]. Lemma

2. Define

(4.2)

+4

F(t)=2

oe

be

S$) @?(n)n7\(t — 20)

d(n)d(n + f)f7* sin (5)

exp (—(f/2nU)?) .

n 1, we have

aX het! if |D(-ae"**)| dia x

oo ax ; oF a(m)a(n) f € (xe~“ (me? = ne~*°)) dx m,n=1

d(m)d(n)

Tite

ss

i

3

|m — n|

m,n cats(f)u - cos (Kj log(K; /4et)) exp (—(K;/2Ut)”) . Finally, summed over f, this amounts to the formula (6.5). The

Dirichlet

series approach.

The zeta-function ¢+(s) was analyzed by

Tahtadjan and Vinogradov [TV] (see also [J3] and [J4]) on the basis of its approximate representability in terms of a certain inner product involving non-

holomorphic Eisenstein and Poincaré series. Recently we sharpened (in [J4]) the estimate for the accuracy of this approximation, and in fact the representation of ¢;(s) can be made perfectly explicit.

In the Tahtadjan-Vinogradov expression of ¢¢(s) as a meromorphic function, there is a leading term with a triple pole at s = 1 giving the main term in the additive divisor problem. Likewise, this pole accounts for the main term for the integral (6.3) when the integration is shifted to the left. The next poles are encountered at the points z; and 2} with z; = $ +1«;. These are poles of the function 1/2-s

1

PT(s/2) 2. lal ty F)-HG (5) (25 /2)[7P(s — 25)0(s — 24), which represents the contribution of the discrete spectrum to ¢(s). The cor-

responding residues of the integrand in (6.3) account for the explicit terms in (6.5). We ignore here the remaining less significant ingredients in the formula for ¢f(s). The term corresponding to the pole z; is

(6.7)

5 leat ts(F (4S)

HS (S)IP(25/2)/*(C (25/2) *T(2in M5;) (z;).

By Stirling’s formula and the definition of aj, we have

losl?IC(25/2)4TRinj) _1 fay 91/8)

Pag

Hier e.

Vragh: [242ins

Further, the saddle point method gives, as above,

v(iM(te/ms 2n )t/ «5 ft/n,2V M,(zj) © —te(1/8) 5 fr

2+ing exp (—(«;/2Ut)?)

FOURTH MOMENT

OF RIEMANN’S ZETA-FUNCTION

533

Thus the expression (6.7) is approximately

T

|

re

ee

aU Ft ei M(t fy Paasts (FHF (5); ? (det /n)**9 exp NeedGry 0k2 The real part of this represents one half of the combined contribution of the poles z; and Z; to our sum over n for fixed f, and the rest of the argument is as above. 7. Remarks

on cusp form L-functions

The key property of the zeta-function in the preceding argument was the functional equation for ¢?(s), hidden in the proof of Lemma 1. Since the gammafactors in the functional equations of the cusp form L-functions are asymptotically similar to those occurring in the functional equation for ¢?(s), an extension of the above method to mean square problems for cusp form L-functions seems plausible. As an example, let us consider the L-function

Hia)ian inn m=1

related to a fixed Maass wave form. Thus the t(n) are the coefficients t;(n) for certain fixed index 7 which we drop from the notation, and likewise we write « in place of k;. Let us suppose that the form u(z) = u(x + yi) in question is

an even function of x. Then it is well known that H(s) satisfies the functional equation H(s) = x(s)H(1 — s) with (see [EHS]) 4

Maser

any

ewe! +1iK — s) T($(1 — ik — 8))

T(E(s + in) P(4(s — ix)

Note that this equals the corresponding function in the functional equation for ¢?(s) if we put formally « = 0. For fixed «, its behaviour is still similar as

| Im s| tends to infinity. Turning to the argument in [T2], §7.13 and 7.16, the role of the function ¢(z) in (1.4) will be played by the series [ee]

d(z) = Ss ine: n=1

There is now nothing to correspond to the second term in (1.4), because H(s), as an entire function, has no pole at s = 1. An analog of (1.3) then holds as such. Further, an analog of the crucial Lemma 1 also holds in a weaker form

with the error term O (|z|!/?~*(m/2 — |arg 2|)~*), which just suffices for our

MATTI JUTILA

534

purposes for we need that the exponent of |z| be less than 1/2. To verify this transformation formula along the lines of Titchmarsh, the relation T(1 — s)

x(s)

= (2n)*~?* (-i + O((lt| + 1)™*)) P(s)

is needed, and this can be seen by Stirling’s formula. In the notation of [T2], §7.16, the parameter a is taken to be 1/2 —«, and 6 stands for 7/2 — |arg z]. A standard mean value estimate for H(s) is also required. The mean square of H(s) on the critical line can now be reduced to sums related to the sum function

T(x,f)= >_ t(n)t(n+

f).

n 1. The zeta-function

2 tnjt(n + f)(n+ f)-* naturally plays the same role as ¢(s) above; for this purpose, certain recent results ( {J4]-[J5]) on this function are required. In this way, the following

analog of (1.7) and the mean value theorem of Good [G2] may be obtained: Theorem

3. For T > 2, we have

Sel

i |H(5 + it)/? d=

log)

+o)

MO),

0

where c,, C2, and the O - constant depend on the Maass wave form in question.

Acknowledgement.

The author is grateful to Prof. A. Ivié and Y. Motohashi for valuable comments and suggestions concerning the present paper. References [Al]

F. V. Atkinson,

[A2]

line, Proc. London Math. Soc. (2) 47 (1941), 174-200. F. V. Atkinson, The mean value of the Riemann zeta-function, Acta

[EHS]

Math. 81 (1949), 353-376. C. Epstein, J. L. Hafner and P. Sarnak, Zeros of L-functions Attached

[E}

to Maass Forms, Math. Z. 190 (1985), 113 -128. T. Estermann, Uber die Darstellung einer Zahl als Differenz von zwei Produkten, J. reine angew. Math. 164 (1931), 173-182.

The mean

value of the zeta-function

on the critical

FOURTH MOMENT

[G1] [G2]

[G3]

(H-B] [In]

[Iv1] [Iv2] [IM]

OF RIEMANN’S ZETA-FUNCTION

535

A. Good,

Cusp Forms and Eigenfunctions of the Laplacian, Math. Ann. 255 (1981), 523-548. A. Good, The square mean of Dirichlet series associated with cusp forms, Mathematika 29 (1982), 278-295. A. Good,.On Various Means Involving the Fourier Coefficients of Cusp Forms, Math. Z. 183 (1983), 95 -129. D. R. Heath-Brown, The fourth power moment of the Riemann zeta function, Proc. London Math. Soc. (3) 38 (1979), 385-422. A. E. Ingham, Some Asymptotic Formulae in the Theory of Numbers, J. London Math. Soc. 2 (1927), 202-208. A. Ivié, Mean Values of the Riemann Zeta-function, Tata Lect. Notes in Math. vol 82, 1992. A. Ivié, On the fourth moment of the Riemann zeta-function (to appear). A. Ivié and Y. Motohashi, On the fourth power moment of the Riemann zeta-function, J. Number Theory 51 (1995), 16—45. H. Iwaniec, Fourier coefficients of cusp forms and the Riemann zetafunction, Séminaire de Théorie des Nombres, Bordeaux 1979/1980, exposé no 18. M. Jutila, Riemann’s zeta-function and the divisor problem, Arkiv Mat.

21 (1983), 75-96. M. Jutila, Riemann’s zeta-function and the divisor problem. Mat. 31 (1993), 61-70.

II, Arkiv

M. Jutila, The Additive Divisor Problem and Exponential Sums, Advances in Number Theory, Clarendon Press, Oxford, 1993, pp. 113 —135. M. Jutila, The additive divisor problem and its analogs for cusp forms.

I, Math. Z. (to appear). M. Jutila, The additive divisor problem and its analogs for cusp forms.

II., Math. Z. (to appear). M. Jutila and Y. Motohashi, Mean value estimates for exponential sums and L-functions: a spectral theoretic approach, J. reine angew. Math. 459 (1995), 61-87. N. V. Kuznetsov, On the mean value of the Hecke series of a cusp form of weight zero (in Russian), Zap. Nauch. Sem. LOMI, AN SSSR 109

(1981), 93-130. N. V. Kuznetsov, Convolution of the Fourier coefficients of the Eisenstein-Maass series (in Russian), Zap. Nauch. Sem. LOMI, AN

[L} [M1]

SSSR 129 (1983), 43-84. N. N. Lebedev, Special functions and their applications, Dover Publications Inc., New York, 1972. Y. Motohashi, An ezplicit formula for the fourth power mean of the

MATTI JUTILA

536

[M2]

Riemann zeta-function, Acta Math. 170 (1993), 181-220. Y. Motohashi, The Binary Additive Divisor Problem, Ann. Sci. l’Ecole

[T1]

Norm. Sup. 27 (1994), 529-572. E. C. Titchmarsh, The mean-value of the zeta-function on the critical line, Proc. London Math. Soc. (2) 27 (1928), 137-150.

[T2]

[TV]

[Z]

E. C. Tichmarsh, The Theory of the Riemann Zeta-function, 2nd ed., Clarendon Press, Oxford, 1986. L. A. Tahtadjan and A. I. Vinogradov, The zeta-function of the addttive divisor problem and the spectral decomposition of the automorphic Laplacian (in Russian), Zap. Nauch. Sem. LOMI, AN SSSR 134 (1984), 84-116. N. Zavorotnyi, On the fourth moment of the Riemann zeta-function (in Russian). Preprint, Computing Center, The Far East Branch of the Academy of Sciences of USSR, Habarovsk, 1986.

Matti Jutila Department of Mathematics University of Turku SF-20500 Turku FINLAND jutilaQsara.cc.utu.fi

On the uniform of Gauss

sums

distribution

and Jacobi sums

Nicholas M. Katz and Zhiyong Zheng}

Dedicated to Heini Halberstam

Abstract. Let F be a finite field with q elements, let UV be a non-trivial additive character of F*, and y , p be the multiplicative characters of F*. We denote by G(W,x) the Gauss sums and by J(x,p) the Jacobi sums. In this paper, we consider the equidistribution properties of the following sequences in the interval [0,1] as g — oo: 1

{5pase (Uxd}

VAVo,

xFxX0;

and

1 {5 at

Su)

XP XO,

POX

=

-

We establish some equidistribution estimates with better error terms for

them. Also we give moment estimates for G(V,x) and J (x, p).

1. Introduction

and statement

of results

Let F, = F be a finite field of characteristic p, with g elements, F, be the prime field of p elements, tr and N be the trace and norm maps from F

to F,. We denote by WY the non-trivial additive character of F’ defined by W(x) = e(tr(x) /p), so that any additive character of F may be written as WV, (x) = V (az), where a € F. Thus, Vo is the trivial additive character. Let x be a multiplicative character of F* extended to F by x (0) = 0, xo be the ~ trivial multiplicative character.

The Gauss sum is defined by

G (Vax) = D> x(a) Va (a). aéF*

1Research partially supported by the National Natural Science Foundation of China

NICHOLAS M. KATZ AND ZHIYONG

538

ZHENG

The Jacobi sum is defined by

Tap) =

x(a) pl =a), acF*

where p is also a multiplicative character of F. If Va # Vo, x # x0,

P# Xo;

p#x_?, it is known that

IG (Wax) |= lJ tx p) | = Va In other words, g~!/2G (W,, x) and q~!/?J (x, p) lie on the unit circle in the complex plane.

It is of interest to consider the equidistribution properties of

the following sequences in the interval [0,1] as ¢ — ov, 1 \= ang G LT

(a,x)

} )

Va

F Vo,

XxX ca X0>

and 1 {5p eI

< Oxo),

xX # Xo;

Here, arg z stands for the argument of purpose of this paper is to establish some above sequences with good error terms. To any real numbers with 0 < 6, < éd2 < 1, and

P#X0,X

2

the complex number z. The main equidistribution estimates for the state our results, let 6; and d2 be let

Sp(51,50) = #{(Vax): Ya # Wo,x # XO (1)

1

and 6; < — argG(W,,x) < 6} 21

and

ms

IS p (51, 52) : = #4(X09) x # X00 # XOX

2

1

and 6; < 5, ate J (xP) S bo}. The main results of this paper are the following. Theorem

(3)

1.

We have

[Sr (61,2) — (51 — 62) (g — 1) (q — 2)| < 7(q—1)d(q—1)logg

and

(4)

|JS (51,52) — (61 — 52) (q — 2) (q — 3)| < 6q°/8

DISTRIBUTION

OF GAUSS SUMS AND JACOBI SUMS

539

Here d(q —1) stands for the number of positive divisors of g — 1. It is known that d(n) is O(n‘) for every € > 0, and that d(n) is not O ((logn)*) for any 6 > 0, cf. Hardy and Wright, Theorems 314 and 315. To prove the above theorem we use the Erd6s-Turan inequality in conjunc-

tion with good information on the moments of Gauss sums and Jacobi sums. For any positive integer n, we define the moments of G(W,, x) and J (x, p) as follows.

Mae

Si Ge) WaFtVo x#x0

and

X#X0 p#XO,x7!

Our original method of estimating both M,, and JM, was based on the theory of Kloosterman sheaves, cf. {Ka-GKM]. Although this theory will be used below in estimating JM,, we recently realized that there is an entirely elementary method of estimating M,. Theorem and q—1.

(5)

2. Let n > 1, Then

d:= gced(n,q —1), the greatest common

divisor ofn

|Mn| < (d-1)(q—-1)q™”.

Theorem

3. For any positive integer n, we have the estimate |JM,,| ze og

a

except in the exceptional case q even and n = 3 orn = 7, in which case we have \JM,,|
I (x%x0)” X#X0

PFO

X#X0

= JM, + (-1)” S> x((-1)") + 2(-1)" (q- 2), X#Xo

which gives

V, +2(q—2)-—1 JMn = < Vn +3(¢ — 2)

V, — 3(q — 2)

if n odd and gq odd, if n odd and q even,

if n even.

So we certainly have the inequality

(14)

|JMn| < |Vn| +3(¢ — 2).

We will prove the following estimate for |V,,|. V, estimate.

For n > 2, we have the estimate

lg"Val

T1,22,""" 5Ln

4)

(Sxteee-e)) (Tea) a

a 2)

p

i)

=(q—1)? | number of solutions in F” of NEF =l= I] (1-= 25) j

j

For any point (t1,t2,... ,tn) € F” we define f (t1,.-. , tn) = number of solutions in F” of ie ==

Ne I] (bat ae

j

j

Thus we get the following lemma. Lemma

2. We have T, = (q - Th ec (ik ee abe

The idea is to view f as a complex valued function on F”, and to compute its Fourier Transform (using the fixed non-trivial additive character V of F to define the Fourier Transform). We will eventually recover T,, by Fourier inversion.

Lemma

3. We have the formula

(PIA) (000

tn) =

oy 19°"

Proof.

70)

fn=1

To see this, it is best to work in the n-fold product ring Ff”, with

546

NICHOLAS M. KATZ AND ZHIYONG

component-wise operations.

Given x = (21, 22,.--

ZHENG

,Zn,) € F”, we define

n

2)

Norm (x) = I] z;,

Trace (x) =

j=1

y Li i

For fixed t = (ti, t2,... ,tn) € F”, f (t) is the number of solutions in F™ of the equation {e+y=t,

Norm

(a) ==

Norms (=

Le

as one sees by writing y := t—2. Let W be the non-trivial additive character of F defined by V (a) = e(tr(a@)/p), and use it to define the Fourier Transform. Then by definition, FTf is the function on F” defined by

a =(47,09,... 565) > (FETA) (ae

Se (do ajts) FO ter”

= a W (Trace (at)) f (t) tern

By the definition of f (t), we have

FTF (a)

oe

W (Trace (a(x + y)))

r,yerr Norm

=

(2)=

Norm

>

(y)=1

WV (Trace (az))

cern Norm(az)=1

CE ¥(Sen)). : £1%9°"-f,=1

For UV # Wo, a € F*, we denote by K1(n, V, a) the Kloosterman sum

Kl(n,¥,a)=

S$)

W(a, +a.+--» +a).

©1%2°''ln=a

Lemma

4.

We have

PEIN OS oO) ,

wd

ap) (PTS) Giyaeenan)

1 ,A2,°"" ,an€F™ a;#0

Ns

Sy

0| Kl(n, V,0)|? KU(n, ¥,c)] < n2qr-2)/2, S c#0

except in the exceptional case when q is even and either which case we have the estimate

Sa

Kl(n, W,c)|*

Kl(n,

V,c)

=< ee

n = 3 or n = 7, in

ont g ry)

a

c#0

Proof. For n = 1, the assertion is trivial, since the sum we are estimating is equal to —1. Thus we may assume n > 2. Pick a prime | # p := characteristic (Ff). The sum

(-1)"* S°| K1(n, ¥,c)|? K1(n, ¥, c) c#0

is the sum

of the traces of Frobenius at all rational points of G,, 7 for the

lisse sheaf Kl(n,V) @ Kl(n,V)@ Kl(n,WV). This sheaf is lisse on Gn of rank n3, pure of weight 3(n — 1), tame at zero, and all oo— breaks are < 1/n.

Suppose we know H? (Gm,r, Kl(n,V¥)@ K1(n,V) @ K1(n,WV)) vanishes in the non-exceptional case. Then the Euler Poincare formula shows that its H2}

is of dimension < n?. By Deligne this H} is mixed of weight < 3n — 2, and we get the asserted estimate. To show that H? (Gm,r, Kl(n,¥) ® Kl(n,¥)® Kl1(n,W)) vanishes in the non-exceptional case, we argue as follows. First of all, up to a Tate twist,

Kl(n,¥) ® Kl(n,W) is just the sheaf End( Kl(n,W)).

Let us denote by

Ggeom the geometric monodromy group of K1(n,W), cf. [Ka, GKM, 11.1] for the determination of Ggeom. We must show that Kl(n, ¥)@End( K1(n, W)) as a representation of Ggeom does not contain the trivial representation of Ggeom. Suppose first that either q is odd, or that n is even.

Denote by Zgeom the

center of the group Geom. It suffices to show that K1(n, V)®End( K1(n, ¥)) as a representation of Zgeom does not contain the trivial representation of Zgeom-

If n is even,

Ggeom is Sp(n),

and Zgeom is the group

+1, which

acts

by its non-trivial character on Kl(n,¥) ® End(Kl(n,W)). If n is odd and q is odd, then Ggeom is SL(n), whose center wz acts as itself on Kl(n,V) @ End( Kl(n,W)). so in both cases, Zgeom acts through a single

NICHOLAS M. KATZ AND ZHIYONG ZHENG

550

nontrivial character of itself on KI(n,¢) @ End(Kl(n, Y)), and hence we get % the asserted vanishing of H?. a Vv) (n, kl case this In odd. is 2 Suppose next that q is even, and that n > V)@ Kl(n, V)® Kl (n, U) (since W takes values in +), and so the sheaf Kl (n, kl (n, W), is just Kl (n, )°?. One knows [KA-GMK, 11.1] that Geom for Kl (n, VW) is given by Ggeom = SO(n)

Geom =

for n odd, #7

the subgroup G2 of SO (7) for

= 7

Denote by std, the standard n-dimensional representation of SO(n). To complete the proof of the vanishing of H? in the non-exceptional case, we must show that for odd n > 5, std,© has no non-zero SO(n)-invariants. Since the standard representation is self-dual, this amounts to the statement that std, does not occur in std,©”. But this is clear for n > 5, since as SO(n) representations we have

(std,)°* =

Sym? (std,) ® A’stdn.

Now A?std, is the adjoint representation Lie(SO(n)) of SO(n), and is irreducible for n > 5 because Lie(SO (n)) is simple for n > 5. As for Sym? (std,), it has the decomposition

Sym? (std,) = 1@

SphHarm?

into the direct sum of the trivial representation and the space of spherical har-

monics of degree 2, which is known to be irreducible. Thus the decomposition into irreducibles of (std,)®? is

(std,)°”

=1@ SphHarm? @ Lie ( SO(n)),

and none of these three can be std,, as already their dimensions are different. This concludes the proof in the non-exceptional cases. It remains to treat the exceptional cases q even, n = 3 or n = 7. In both these cases, we first show that the H? in question is one-dimensional. For n = 3, Ggeom is SO(3), and the decomposition into irreducibles of (std3)°? is

(stds)°? =1@

SphHarm? @ Lie( SO (3)).

Here SphHarm’ is five-dimensional, so certainly not std3, but for n = Lie(SO (3)) is the standard representation of SO(3). Thus we get

(std3)®?

=1@ SphHarm? 9 std.

3

DISTRIBUTION

OF GAUSS SUMS AND JACOBI SUMS

501

and hence (std3)®* has a one-dimensional space of SO(3)-invariants, as required. (Using autoduality of std3, one can “see” this invariant as the determinant, an interpretation we do not need here.) We now turn to the case n = 7. Here Ggeom is the subgroup G2 of SO(7). As representations of SO(7), we have a decomposition into irreducibles

(std7)®”

=1@ SphHarm? @ Lie (SO (7)),

with dimensions 1, 27, 21. Under G2, SphHarm? remains irreducible (the dimension of the irreducible representation of G2 of highest weight 2w is 27, as one computes using the Wey] dimension formula, cf. Bourbaki, Lie VIII, §9, n° 2 , page 152, where it is given explicitly for G2). But under G2, Lie (SO (7)) decomposes as Lie (SO (7)) |G2 =

Lie (G2) ® std7.

So all in all, the decomposition into irreducibles of (atd;)> 1Gs is

(std7)®? =1@ SphHarm? @ Lie (G2) @ stdz, of dimensions

1, 27, 14, 7.

Hence

(std7)°? has a one-dimensional

space of

G»-invariants, as required. Recall that we are trying to estimate, for g even and n = 3 or n = 7, the sum

(-1)"* >| Kl (n, ¥,¢) |? K1(n, ¥,c), cA#0

which, because WV is +1-valued for g even, we may rewrite as

Le

iG vcs c#0

the sum of the traces of Frobenius at all rational points of Gr for the lisse sheaf K1(n, wy)? . We have just proven that for n = 3 or n = 7, this sheaf, on G,,7, contains one copy of the constant sheaf. Therefore, at least one of its oo—slopes must vanish. But as all oo—slopes of Kl(n,WY) are 1/n, all oo—slopes of Kl(n, yw)? are 0, and if g(n) < G(n), then (3.8)

V (G(1)}

yee)

Vp—a(klBn + 4),

ESTIMATION

OF EXPONENTIAL

SUMS

579

for some real y, and

(3.10)

ENF)

aig ss

Vip)

hi

= Dy.

To prove Proposition A, we also need the following two lemmas.

Lemma 3.1. Let g(n) be a real arithmetic function, f(n) a polynomial with real coefficients, and

S=

J> g(nje(f(n)). a > (P(BR))™® =(y = m1 Bi + +++ + re Be ET). Observe that the subspaces V° and V+ of V, generated by the subsets [° = {y €T : ¥(y) = 0} and I+ = {y ET: ¥(y) F 9}, satisfy V° © v= Hence the preceding construction of B splits up in two parts, that of a basis B* = {f;,...,(%} of V+, and that of a basis B° = {6:41,..., 84} of V°, say. We set

p(B) =0

for

BEB,

and are left with the system of equations

b(y) = (p(A1))™ + (PCB)

(Y= 1B ++

+ me EP).

Writing ¥(y) = e”™ and (8) = e?) with w(y7), z(B) mod 27i, we obtain the system of linear equations wy) = 12z(B1) +--+ 42z(F:)

mod

2mt

(y=mfi+---+ ne

€ ad)

for z(3;),..., 2(). It reduces to a system of ¢ linear equations having rank t, since w is multiplicative on P+ C A and dim I+ = ¢. Therefore its determinant is a nonzero integer +D with D € N, and the numbers z(/),..., z(G) are uniquely determined mod a which shows the existence of a nontrivial multiplicative homomorphism y : M — C coinciding with 7 on I. Observe that, in particular, the vector space V over Q generated by I contains M and, by construction, has minimal dimension. Hence, in this situation, the boundedness of w implies that of y. We now turn to the case r > 1. Consider the projections [),...,T,. R, of [CR,” in the directions €1,--.,€,, representing the vectors of the standard basis of the real vector space R". By applying the one-dimensional version of Lemma 1 to each of the finite sets [y,...,I, with respect to the corresponding projections

A;,...,A,

of A, the existence of the sets Bigs:

Dey

WEIGHTED

WIENER-LEVY THEOREMS

615

having the desired properties follows. This construction of B,...,B, gives vector spaces V; D [j,...,V, D> I, of minimal dimension over Q, and M = {B,e, + -:- + Be, | Bi € Bi,...,8, € B,}. Hence the existence of a

nontrivial (bounded) homomorphism y : M — C such that y = # onT

follows

as in the one-dimensional case. Returning to the proof of the Density Lemma we notice that Lemma 1 associates the finite set [ C A occurring in (12) with a finite set T Cc N¢. Namely, let By = {Go1,---, Pox, } where kp € No for g=1,...,r in Lemma 1, and set k = kj +---+k,. We may assume k € N. Then T is defined by

(jay keys

oe

sty eee pie

if and only if

A = (n11 811 +++

+ 1b, Pik) €1 + ++

+ (Mri Ber +++ + Ark, Brk,) er ET.

Furthermore, Lemma 1 carries over the multiplicativity and the boundedness by 1 from w to y. This gives

P(A) = (p(Gr1e1))"* ++ (P(Bre,er))"** and the sums occurring in (12) take the form

Yo eQ)¥OA) = Q(~(Bire1), «++ -P(Brkxer)) € QD") der with a polynomial Q in k complex variables, and similarly

So a(a)en>* = Qe, ... ePrtrPr)€ Q(n(H")), AET

where 77 : H’ — D’ is defined by

(15)

z=n(s) =(eP"",... Bp

a ae ee

hc)a)

fois = (Spec cseen) Density Lemma.

Hi’. Finally, the next lemma completes the proof of the

Lemma

that f : ioe — C is a continuous function, holomorphic

on D*.

2. Assume

Then, with the mapping 7 : ees De from (15), the set f(n(H’)) is

dense in f(D").

Proof. We adapt the method of Schwarz and Spilker [15], where the case r = 1 is treated as Hilfssatz 5.1. Let vu € f(D") and g(z) = f(z) —v for z€ D*. We

616

LUTZ LUCHT AND KLAUS REIFENRATH

have to show that

(16)

inf{|9(z)| | z € n(H”)} =0.

Suppose, on the contrary, that with some ¥ > 0

(17)

lg(z)| > 0

For fixed o = (o1,...,0,)

forall z €n(H’).

€ Ry we set

Loe ={zeE Cra Zon| = enPen?e (1S Kee | RSS Since By, =

Se NP

{Box | 1 < & < ko} C Ry is linearly independent

over Q for

every 0 =1,...,7, the set n({o + it | t € R"}) is dense in L, by Kronecker’s approximation theorem (see, for example, Hardy and Wright [4], Theorem 444). Therefore (17) even holds for all z € L, with ao € R,”. Since 0 € D* is limit point of n(H'’) we obtain also |9(0\ | from. (17). For fixed ¢ = (o1,...,0,) € Ry we set further K, ={zeC*

| IZon| S ear tea (Tee

ko l 0 by

— 0 a =inf{é € Ry |\g(z)| 25 for allz € Ke}. Since g is a zerofree continuous function of K,, and holomorphic on Kg, for every 3 > a, the maximum principle yields

max {|7

eS Ko} = max {||

2 € Ins}
8

for all z € Ka.

Assume that a > 0. Then |g(z)| < J for some z € Kg, by definition of a. Hence a = 0 and

lg(z)|>9

forallzeD*,

WEIGHTED

WIENER-LEVY

THEOREMS

which contradicts v € f(D*) and proves (16).

617

This completes the proof of

Lemma2.

4. Inversion

theorems

and Euler products

The case F(z) = z~*, Q = C% of Theorem 2 is of specific interest. Evidently (4) is then equivalent to the Wiener type inversion condition

(18)

la(s)| > 6 >0 for alls € H”.

An important class of functions a € £1,(M) is that of multiplicative functions in which case @(s) has an Euler product representation. In order to explain these notations we need some arithmetical preparation. It is well known that the Euler product representation of an ordinary Dirichlet series having multiplicative coefficients is the analytic analogue of the Unique Factorization Theorem in N. In this section, we consider free multiplicative semigroups M of real numbers n > 1 with 1 € M, by which we mean that there exists a countable subset P C M of free generators, the “prime elements”, such that every n € M, n # 1, has a unique decomposition into finitely many elements of P, apart from their order. Observe that, in contrast to the notion of a multiplicative arithmetical semigroup in the sense of

Knopfmacher [8], M may have finite limit points. If M),..., M, are free multiplicative semigroups generated by P;,...,P,, then M, x --- x M, is generated

by

UGbataes

oo spo con a

1l 0. Hence, with

=> 1 |[%>9, pEG

(18) results.

References

[1] Edwards, D. A., On absolutely convergént Dirichlet series, Proc. Math. Soc. 8 (1957), 1067-1074. [2] Edwards, R. E., Fourier series, Vol. I and II, New York 1967.

Amer.

WEIGHTED

WIENER-LEVY

THEOREMS

619

Gelfand, I. M., Uber absolut konvergente trigonometrische Rethen und In-

tegrale, Rec. Math. (Mat. Sbornik) N.S. 9 (1941), 51-66. Hardy, G. H., Wright, E. M., An introduction 3rd ed., Oxford 1954.

to the theory of numbers,

Heppner, E., Schwarz, W., Benachbarte multiplikative Funktionen, in: Studies in Pure Mathematics (To the Memory of Paul Turdn), Akadémiai Kiad6-Birkhauser 1983, 323-336.

[7]

Hewitt, E., Williamson, J. H., Note on series, Proc. Amer. Math. Soc. 8 (1957), Ingham, A. E., On absolutely convergent Math. Analysis and Related Topics. Essays 1962, 156-164.

absolutely convergent Dirichlet 863-868. Dirichlet series, in: Studies in in Honor of G. Pélya, Stanford

[8] [9]

Knopfmacher, J., Abstract analytic number theory, Amsterdam 1990. Lévy, P., Sur la convergence absolue des séries de Fourier, C. R. Acad.

[10]

Sci. Paris 196 (1933), 463-464; Compositio Math. 1 (1934), 1-14. Lucht, L., An application of Banach algebra techniques to multiplicative

functions, Math. Z. 214 (1993), 287-295.

[11] [12] [13] [14] [15]

[16] [17]

Lucht, L., Weighted relationship theorems and Ramanwan expansions, Acta Arith. 70 (1995), 25-42. Newman, D. J., A simple proof of Wiener’s 1/f theorem, Proc. Amer. Math. Soc. 48 (1975), 264-265. Reifenrath, K., Gewichtete Wiener—Lévy-Satze und arithmetische Halbgruppen, Dissertation, TU Clausthal 1995. Rudin, W., Functional analysis, New York 1974. Schwarz, W. Spilker, J.. Wiener-Lévy-Satze fur absolut konvergente Reihen, Archiv Math. 32 (1979), 267-275. Wiener, N., Tauberian theorems, Annals of Math. 33 (1932), 1-100. Wiener, N., The Fourier integral and certain of its applications, Cambridge

1933.

Lutz Lucht and Klaus Reifenrath Institut fiir Mathematik

Technische Universitat Clausthal Erzstrage 1 - 38678 Clausthal—Zellerfeld

GERMANY

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rooney

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;

vedio’

_ 7

ia

ae ser 0 for each 7, and it has the Fourier expansion y ==—J) T'(s;)

(2a |nly)e(nz). U5 (nr) ( ) Ki, t; ( | ly) ( )

The Fourier coefficients v;(n) are proportional to the eigenvalues A;(n) of T;, :

vj(n) = 9; (1)Aj(n),

form > 1.

The eigenvalues A;(n) enjoy the following multiplicative property:

mn AyOm)As(n) = > Ay (A) d|(m,n)

The automorphic function associated to u;(z) is given by the absolutely convergent series co

Bia

pyar n=)

for Rs > 1, which has analytic continuation to an entire function and satisfies one of the functional equations 6;(8)H;(s)=6;(1

— s)HF(1

—s),

if 6; =

O;(s + INH; (s) = —0,(2 — s)7i,(1— 3),

in

“if 0; = =,

where

Our main result is

Theorem

1. We have

(1)

a

[H;(s;)|*

Pee

eke

T1.

The implied constant depends only upon e.

We have H;(sj)

« t87*.

Theorem 1 will be deduced from the following general inequality:

SPECTRAL Theorem

(3)

MEAN-VALUES 2.

OF AUTOMORPHIC L-FUNCTIONS

623

We have

DS

LDS andy(nm)? < (N+ T)(NT)*

T 1, (13) E(z,s) Es y? os jae

=

5)¢(2s ed 1)

13

D'(s)¢(2s) Meno + Oaey

pits ?01~-26(|n|)K,_1(2n|n|y) exp(27inz) .

Here ¢(s) is the Riemann zeta-function, o,(n) the sum of the ath powers of the divisors of n, and K,(x) the K-Bessel function of order v (cf. [3, p.140]). Thus E(z,s) is a meromorphic function over the entire s-plane.

ON KUZNETSOV’S

TRACE FORMULAE

643

We then consider the Petersson inner product of P(z,s1) and P,(z, 52) with m,n > 1 and Re(s;) = 0; > $j == is

(Pts

$2), Pn (-5 88) =

| Pale

s)Pae

abdul),

where ¥ is the usual fundamental region of [, and du(z) Poincaré metric. The estimate (1.4) implies that the integral lutely and represents a regular function of s; and s» in the As a particular instance of the unfolding argument of Rankin see that the expansion (1.2) gives, for 01,02 > 1,

= y~?dxdy the converges absoindicated range. and Selberg we

1)(4mm)1~81—82 (Pm(+, $1), Pr(-,52)) = Omnl (81+ $2 —

As

a yer =

tSTM

S82—8,—-1l1

=

17251 5

S(m,n50)

-] 5

; ap~ eae Danae

where dm,7 is the Kronecker symbol.

21 esa

dێ ee om ea

To ensure the absolute convergence

on

the right side we impose the condition (1.7)

CoO

la

In it we may exchange freely the order of sums and integrals in (1.6) and have (Pec

Si) piki e,8 2) Tai cme

(1.8)

Gh a 89)

D(Amm)\n

ae Se [7251 S(m,n;l) if (1 ais ca mee Sear (€ ;m,n,l)d€,

1=1

Ee

where oo 2 Y,,(€;m,n, 1) = ik aes exp (— 2nny(1 + 2€) — mae)

So far we have followed the Selberg-Kuznetsov argument. But we now depart from it. We are going to transform the integral in (1.8) into an expression similar to Barnes’ integral representation of the hypergeometric function. To this end we note first that Mellin’s formula gives Yoke pind!)

2am = hyLisl fa bet expl—2any(t +18) (esl once

7

dindy,

YOICHI MOTOHASHI

644

where (a) is the vertical line Re(7) = a > 0, and |Arg(1 — i€)| < $m. This double integral converges absolutely if Re(w) +a > 0. On it we exchange the order of integration and compute the inner integral, getting

: 2 =2n (1s YolEimimt) = seeeany fFroNRn ew)(7) eee, where |Arg(1 + 7£)| < $7. So, providing Cia

a+o2-—0,>0,

we have [are

i hh lane ratae m, 7 ay ae

py any.

I

Oe

erst

(1 + 2€)0+82

This double integral converges absolutely if

(1.10)

a-o,+5 3 and

O(s1,52;7T) =T(s1 - 5 +ir)T(s, — 5 —ir)T(s2 - 5 + ir)T(s2 — 5 — ir). It should be remarked here that as is shown uniformly for any integer m > 1,

(1.19)

>

in the next section

we have,

|as(m)Pe-**!

Iaj(m)Pe™ 2

—TK;

K/2 1. This suggests apparently the relation lo)

(3.6)

w(t

ei =sinh(nt)

We are going to prove it rigorously.

oo cosh(mr)

g(t, 7) Join (x) dr.

ON KUZNETSOV’S

TRACE FORMULAE

653

To this end we consider the expression

*(t,n) =< a sinh(at

37 (

)

-

( 7)

t >

a

[(n — 3 +ir) ae rBEA aa oe EY )i

éosh(nr)

Z r(3 Sits

p]

ir)

where |Im(t)| < a+ 4 < a (cf. (2.4)). We shift the contour to the line Im(r) = —K with an arbitrary integer K > 1. We can sum the residues thus resulting by the formula (3.3), and find that

ons

sin(77) n

T(n + it) 0(n — it) + wk (t, 7)

Mg i 2t cosh(mt)

(kK +it) T(K+7+ it) n T(K+1-7n+ it)

_ (Kit) TK +n-2t) "hd 00, Geer etn where w3(t,7) is the contribution of the integral with the new contour.

By

Stirling’s formula we see that the expression in the braces is O(K?%~') and also

widen) < f+ he co

*

K

—2n|r|

T(K+n-$+i (K+7-35 ir) |r

=00:


3 and @ is as in (1.18).

Then we specialize (5.2) and (5.4) by (2.1), getting

co

—on on wa

j=l

egypt + airy (mn) |C(1l 5 ; Ne Se

BOBS)

ha

= = wh 7S(on, —n;l)w_ (t,=

29

er

)

(ul

where g(t,7r) is as in (2.2), and

i

w_(t,2) = —

271

/ D(n + 2t)0(n - it)(=)

(a)

z

1-2

Gp

with t, @ satisfying (2.4). As before we let f satisfy C, multiply both sides of (5.5) by cosh(mt) f(t + $i), and integrate with respect to ¢ over the real axis; here the absolute convergence is easy to check. Then we are left with the evaluation of

iesw_(t, x) cosh(mt) f(t+ 5i)dt. —oco

It is equal to

(5.6)

=

(=)

1-—2n

oe

/

P(n + it)P(n — it) cosh(at) f(t + 5i)dtdn.

ON KUZNETSOV’S

TRACE FORMULAE

663

In the inner integral we shift the contour to Im(t) = —3, and it becomes

-if- D(n + it+ $)0(n - it — 5) sinh(xt) f (t)dt

= =f

{ries $)I(n — it — 3)

—T(n —it + $)P(n + it - »|sinh(mt) f(t)dt = / i(n + it — $)0(n — it — $) sinh(xt) f(t)dt. We insert this into (5.6), exchange the order of integration, and compute the inner integral by (5.3). We thus obtain the following version of the Kuznetsov-Bruggemann trace formula: Theorem

3. /f f satisfies C, then we have, for any integers m,n > 1,

Bee H+ ef Gane ayes BS simone

=

(mY

where

vay = 4 [_rsint(ar) Kail) f(r)

(5.7)

Now we turn to (5.1) and assume that ¢ satisfies the above condition D. Unfortunately we do not have an analogue of (4.2), which would make our problem easier. We can, however, come close to it. This is by virtue of the fact that

(5.8)

W_(z38,8+5) =V7e*,

which is equivalent to the identity K1 Ce

(1/2x)2e7*.

The specialization

of (5.2) and (5.4) that is indicated by (5.8) gives, for Re(s) > 3, 1

(5.9)

(21/mn)*s—* Se pas 3(™ —n;l) exp (= l=1 —

pj (™m)pj(—)

2 ~cosh(mes) . 7s

1

[° —

a

i

orir(m)orir(n) s,r)dr "4? _ ie (mn)'"|¢(1 + Dir) ane

664

YOICHI MOTOHASHI

A(s

1 O(s,s+4; rs perSneieey Ar eh

T(2s — 3)

2/0

= 23-48\/m cosh(1r)

— 12s — ir Ie [T'(2s —1+4+ 2ir)P(

oa =

(

We note, in advance, that we have

Nar) = 2eosh(ar) f Beek

(5.10)

(cf. (1.15) and (1.16)). We are going to extract the sum (5.1) from (5.9) by integrating the latter. But the presence of the exponential function on the left of (5.9) causes a minor trouble. To overcome it we introduce a C™ function wx depending on a positive parameter X such that Dy wx

for aA,

(x) =

OSStoreg 2a22Xe: and for each fixed v > 0

(5.11)

a)(2) 4m\/mn which we suppose hereafter. We follow closely the argument that we used to prove (4.9). Then we see that the formula (5.9) gives

(5.12) K_(m,n;y) =

$ palem)pj(~n)

A)

ay

es —oo(mn)" |¢(1 + 2ir)|? Ax (r)dr,

=|

ON KUZNETSOV’S

TRACE FORMULAE

665

where

Ax(r).= 1 2

271

A(s,r)0%(s)ds (a)

with a as in (4.8). But (5.10) gives obviously

Ax(r) = 2eosh(ar) |0 oR cx (x) ple)Kain2) dz=. So we have, by (5.3),

eo (r) —Ax(r) = deosh(ar) f° (1— ex c))ola) Kaie(a)= Za cosh id T(s= (ee #ho ir)(s Fea 1

= —cosh(mr

—$—ir

*(s)ds, (1 l-w — wx)p)*(s)

where y~ is defined by (5.13), @ > } and

(A= @x)9)*(s) = /“(d= wx(2))e(@)(2) ae. The condition D and (5.11) imply that ((1 - wx))*(s) is regular for Re(s) >

— 6 and there

(1 — wx)y)*(s) « X~29(|s|

+1).

Thus we have 1 yp Saree (r) — Ax(r) 4 ae ee cosh(ar)

f

ae) s

+ cxh(nrf (1 — wx)y)*(F + ir)P(2ir)

+(l—axy)eG— wpr-ain which is O(X~79(1 + |r|)~2~2°). We insert this into (5.12) and let X tend to _

infinity. In this way we are led to

YOICHI MOTOHASHI

666 Theorem

4.

[f ¢ satisfies the above condition D, then we have, for any

integers m,n = 1,

5(m,—n;)o(2¥™)

ioe||)— ~

1

2

1

[°°

oair(m)orir(n)

p;(™m)p;(—n) *~ (K;) a ae in (mn)"|¢(1 = ~cosh(mK;) = ye

n

a 2ir)|? Yp

(r)dr,

where

(5.13)

a(t) = 2cosh(nr) | (2) K2ir(2) —.

Acknowledgement. We are indebted to Professors H. Iwaniec and M. Jutila for their valuable comments which they made on the draft of the present paper. We note that after finishing this work we learnt that Professor Iwaniec recently published his lecture notes “/ntroduction to the Spectral Theory of Automorphic Forms” ( Revista Matematica Iberoamericana, Madrid 1995 ). In this elegant book a unified treatment of the spectral resolution of the hyperbolic Laplacian as well as the subject related to ours can be found.

Addendum (Feb. 26, 1996): In the meantime we have obtained a further simplification in the proof of Theorem 1. Thus the identity (3.2) can be dispensed with, and we can proceed directly from (2.2) to (3.10). Also we have found that the above argument can be refined to yield a unified account of the trace formulas and the Fourier expansion of the resolvent kernel of a Fuchsian group, which is based on the theory of real analytic

Poincaré series, and which is quite different from the Niebur-Fay analysis. It may be worth remarking that in our new argument for the resolvent the following integral formulas of the Barnes type play important réles:

=f gsin(ne)r(s: +60 (s2 + OT (65 +6) (T)

“D(a = €) (sa =

SE sate dé

T(s; + 5)D(s2 = $)0(s83 ar 5) P(s1 sr 82)T(s1 AF $3):

-T(s2 + 83)/T'(s1 + 82 +8345),

ser | §sin(2ne)P(s1 + (2 + OE (s9 +6) -T(s1 — €)0' (82— €)0'(83 — €)d€ = [(s1 + s2)I'(s1 + 53)T'(s2 + 53).

ON KUZNETSOV’S

TRACE FORMULAE

667

Here the paths separate the poles of ['(s; + €)I'(s2 + €)I'(s3 + €) and those of I(s; — €)P'(s2 — €)I(s3 — €) to the left and the right, respectively; and the parameters $1, 52,53 are assumed to be such that the paths can be drawn. The

proof of these formulas is not difficult. Using (+) we may transform (1.11) to an identity that is equivalent to the result of applying Theorem 1 to the right side of (1.18). In the same way the formula (t) yields the transformation of (5.2) that is equivalent to the result of applying Theorem 3 to the right side of (5.4). The point is that we are able to do these transformations without appealing to the spectral resolution of the hyperbolic Laplacian. The new Fourier expansion of the Poincaré series thus obtained is essentially equivalent to the Fourier expansion of the resolvent kernel for the hyperbolic Laplacian over the full modular group. Naturally the argument extends to any co-finite groups. The details will be published elsewhere.

References

[1] R.W. Bruggemann, Fourier coefficients of cusp forms, Invent. math., 5(1978), 1-18. [2} N. V. Kuznetsov, Petersson’s conjecture for forms of zero weight, and Linnik’s

conjecture,

Preprint,

Habarovsk Complex Res. Inst., East (Russian); see also Math. USSR

Siberian Branch Acad. Sci. USSR, 1977.

Sbornik, 39(1981), 299-342.

[3] N. N. Lebedev,

Special Functions & their Applications,

Dover Publi-

cations, New York, 1972.

[4]

Y. Motohashi, On the Kloosterman-sum Acad., Ser.A 71(1995), 69-71.

zeta-function,

Proc. Japan

[5] A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Symp. Pure Math., 8(1965), 1-15.

Yoichi Motohashi Department of Mathematics College of Science and Technology Nihon University Surugadai Tokyo-101 JAPAN [email protected]

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