Prime geodesic theorem

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

Y

INGCHUN

C

AI

Prime geodesic theorem

Journal de Théorie des Nombres de Bordeaux, tome

14, n

o

_{1 (2002),}

p. 59-72

<http://www.numdam.org/item?id=JTNB_2002__14_1_59_0>

© Université Bordeaux 1, 2002, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Prime

_geodesic

Theorem

par YINGCHUN CAI

RÉSUMÉ. Soit 0393 =

PSL(2, Z).

On démontre _que

03C00393(x)

= lix ₊

&#x3E;

0,

où

_l’exposant

améliore

précé-demment obtenu _par W. Z. Luo et P. Sarnak.

ABSTRACT. Let 0393 denote the modular _group

_{PSL(2, Z).}

In this

paper it is

proved

that

03C00393(x) =

lix + &#x3E; 0. The

exponent

improves

the obtained

_by

W. Z. Luo

and P. Sarnak.

1. Introduction

Let r denote the modular _group

_{PSL(2, Z).}

_By

definition an element

P E r is

_hyperbolic

if as a linear fractional transformation

-’ 2013

/

it has two distinct real fixed

_points.

_By

a

_conjugation

_any

_hyperbolic

element P can be

_given

in a form P =

0’-1 P’O’

with a E

_SL(2,

_R)

and

P’ =

t 0 t

&#x3E; 1. Here P’ acts as

multiplication

P’z =

t2z.

The

factor t2

is called the norm of

P,

let us denote it

_by

_NP,

it

depends

only

on the class

{P}

of elements

_conjugate

to

_P,

NP =

N{P} -

t2.

P and

_{P}

are called

primitive

if

they

are not essential _powers of other

hyperbolic

elements and classes

_{respectively.}

For

_primitive

P the norms can

be viewed as

"pseudoprimes", they

have the same

asymptotic

distribution as the rational

primes,

where

The

_problem

of

_finding

a formula with

_good

error term was

intensively

studied

by

many

mathematicians,

first of all

by

H. Huber

[4],

D.

Hejhal

[2,

Manuscrit _reçu le 16 _janvier 2001.

Project supported by The National Natural Science Fundation of China _{(grant no.19801021),}

3],

A. B. Venkov

_[13]

and N. V. Kuznetzov

_[7]

before the

_eighties,

not

_always

for the same _{group. The result} were of the

_type

this result was also known to A.

_Selberg

and P. Sarnak

_[12]

_gave a "direct"

proof

of it.

In order to

_investigate

the

_asymptotic

distribution A.

_Selberg

in-troduced the

_Selberg

zeta-function which mimics the classical zeta-function of Riemann in various

_aspects.

The

_Selberg

zeta-function is defined

_by

for

_Re(s)

&#x3E; 1 where

_{P}

runs over the set of all

_primitive

_hyperbolic

classes of

_conjugate

elements in r. The most

_fascinating

_property

of the

_Selberg

zeta-function

_Z(s)

is that the

_analogue

of the Riemann

_hypothesis

is true. In view of this

_property

one should

_expect

an error term

0(x2+~).

Let us

explain why

this result is not obvious. It is convenient to

_speak

of the allied

sum

where ~1P =

log

NP if

{P}

is _{a power} of _a

primitive

hyperbolic

class and

AP = 0 otherwise. Then like in the

_theory

of rational

_primes,

_we have the

following

explicit

formula: Lemma 1

_([5]).

where Sj

= 2

runs over the zeroes

of

Z(x)

on

Re(s)

= 2

counted with

their

_{multiplicities.}

s.

1

Here,

in the sum each

term %

has the

order §j

but the number of

J &#x3E;

terms is

_(refer

to

_[2])

On

_taking

the

_optimal

value T = we obtain the error term

0(x 4

log x).

At this

_point

the situation differs _very much from the one

con-cerning

rational

_primes

because the

_Selberg

zeta-function

_Z(x)

has much

more zeroes than does the Riemann

(( s ).

From the above

_arguments

one sees that in order to reduce the

_exponent

3

one cannot

_simply

handle the sum over the zeroes in

(1.1)

by summing

up the terms with absolute values but a

significant

concellation of terms

must be taken into account.

_Only

after N. V. Kuznetzov

_[8]

_published

his summation formula does this

_suggestion

become realizable. In 1983 H. Iwaniec

_[5]

realized such a treatment and

_proved

that

by

means of Kuznetzov trace formula

_incorporated

with estimates for sums

of real character of a

special

_type.

More

precisely,

the basic

ingredients

in

Iwaniec’s

_arguments

is the

_following

mean value estimate for the Rankin

zeta-function:

. - 1 1.

where

_Rj(s)

is the Rankin zeta-function and

_Re(s)

_{= 1;}

and the estimate for the

_following

sum

where

_(-*)

denotes the Jacobi’s

_symbol.

In the meantime P. G.

_Gallagher

obtained

In 1994 W. Z. Luo and P. Sarnak

_[9]

_proved

Iwaniec’s mean value

con-jecture :

(1.2)

holds with the

_{exponent 2}

_{replaced by}

2 + ~.

By

this mean

value estimate and A. Weil’s _upper bound for Kloosterman _sum,

they proved

for the modular _{group r} =

PSL(2, Z).

In 1994 W. Z.

_Luo,

Z. Rudnick and P. Sarnak

_[10]

made considerable advance in the research of

_Selberg’s

_eigenvalue

_conjecture.

_They

showed

that for _{any congruence}

_subgroup

r c

_{SL(2, Z)}

the least nonzero

_eigenvalue

~ ~

As a

by-product

they

obtained

for _{any congruence}

_subgroup

r C

_{SL(2, Z).}

In this _paper we insert

Burgess’

bound for the character sum estimate

and the mean value estimate

( 1.2)

for the Rankin zeta-function into the

arguments

of Iwaniec and obtain the

_following

result. Theorem. For the modular _group r =

PSL(2, Z),

2. Some

_preliminary

lemmas

Lemma 2

_([9]).

Let

_pj(n)

denote the n-th Fourier

_{coefficients for}

the Fourier

_expansion

at 00

of

the

j-th

Maass _cusp

form,

and

denote the

_{Rankin-Selberg L-function for}

the

_j-th

Maass _cusp

_form.

Then

Lemma 3

_([1]).

_{If q is}

not a _square then

where

_~~~

denotes the Jacobi’s

_symbol

and the constant

_implied

in «

de-pends

at most.

We infer from Lemma 3 a

simple corollary.

Lemma 4. Let 1

_R,

_{R2, a &#x3E; 1, q &#x3E;}

1. Then

Define

where

6mn is

the Kronecker’s delta

_symbol.

Proof.

This is Kuznetzov Trace Formula. See Theorem 9.5 in

_[6].

0

3. A mean value theorem for Fourier coefficients

Let

where

_h(~)

is a smooth function

supported

in

(N, 2N~

such that

By

Rankin

_[11]

we know that

C(2s)Rj(s)

has

_meromorphic

con-tinuation onto the whole

_complex

with

_only

a

simple

pole

at s = 1 with residue 2 cosh

_1rtj.

And

_by

_[11]

we know that

_Rj

(s)

is of

_{polynomial growth}

Lemma 6. We have

with

Proof.

Consider the Mellin transformation

for s = 0’ + zT

_{by partial integration}

1999 times.

_By

the inverse Mellin transformation and

_Cauchy’s

theorem we have

and Lemma 6 follows from Lemma 2.

4. A mean value theorem for

p(c, a)

Let

_{p(c, a)}

stand for the number of solutions

_d(modc)

of

and

where

_cp(c)

is the Euler’s function. Now

Proof.

Write c = kt where k is a

squarefree

odd number and 41 is a

square-full number

coprime

with ~.

_By

the

_{multiplicativity}

of

_{p(c, a)}

in c we

_get

d2 -

ad + 1 -

_0(modc)

in

_d(modk)

is

_equivalent

to

a2 -

_4(modk)

in

x(modk)

and the number of

_incongruent

solutions of the latter is

Let

_Q

stands for the set of

_squarefull

numbers. Then

By

(44)

in

_[5]

we have

with some absolute constant A.

By

Lemma

_4,

after

_{splitting up}

the summation over r in

Foo(A,

B,

C)

into intervals of the form

_(Rl,

_R2)

with

1 - 1 R

R1

R2

2R1

we deduce

Since for _p &#x3E;

_{2, a &#x3E;}

2 we have

p(pc, a) :5

where

pI3

=

a2 -

4,p~)

it follows that

and

_finally

By

(4.3)

and

_(4.4)

with R = we

_get

that

Comparing

(4.1)

and

_(4.5)

for B =

A2, C

= _{o0 one} finds A =

7r2 6

i

which

_completes

the

_proof

of Lemma 7. 0

5. An

_application

of Kuznetzov trace formula Let

Then we have _cp =

p B + WH- It is easy to show that

Let

where

_S(n,

_c)

=

S(n,

_n;

c).

Then

by

Lemma 6 we have

1 i 1

By

Jo(y)

G and we

_get

By

the above

_arguments

and Kuznetzov Trace formula we

_get

6. An estimation for

_En

Since we have Notice that thus Moreover we have

Now

falls into one of at most

_O(logX)

_partial

sums of the form

By

Poisson summation formula we have

Hence

the

_integration

of M over x is

with I =

[1]

₊

_2,

since

ANC! 1 &#x3E;

XI.

Integration

over x from

C )

of R

_yields

Combining

all the above

_arguments

_(6.1)-(6.7)

we

_get

7. Proof of the theorem

By

(5.1)

and

_(6.8)

we

_get

By

(7.1)

and the Fourier

_Technique

used in

_[9]

we

_get

By

(58)

in

_[9]

and the theorem follows from

_(7.4),

Lemma 1 and summation

_by

_parts.

References

[1] D. _BURGESS, The character sums estimate with r = 3. J. London. Math. _{Soc.(2) 33} (1986),

219-226.

[2] D. _HEJHAL, The _Selberg Trace Formula _{for PSL(2, R),} I. Lecture Notes in Math. _548, Berlin-New _York, 1976.

[3] D. _HEJHAL, The _Selberg Trace Formula and the Riemann Zeta-function. Duke Math. J. 43

(1976), 441-482.

[4] H. HUBER, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen

II. Math. Ann. 142 _(1961), 385-398 and 143 _(1961), 463-464.

[5] H. IWANIEC, Prime _geodesic theorem. J. Reine Angew. Math. 349 _(1984), 136-159.

[6] H. IWANIEC, Introduction to the _{Spectral Theory of Automorphic} Forms. Biblioteca de la Revista Matemática _{Iberoamericana,} 1995

[7] N. V. _KUZNETZOV, An arithmetical _{form of} the _Selberg Trace _formula and the distribution

of the norms _{of primitive hyperbolic} classes of the modular _group. Akad. Nauk. _SSSR.,

Khabarovsk, 1978.

[8] N. V. _KUZNETZOV, The Petersson conjecture for cusp forms of weight zero and the Linnik

conjecture. Sums _of Kloosterman sums _(Russian). Mat. Sb. _{(N.S.) 111(153) (1980),} no. _3,

334-383, 479.

[9] Z. Luo, P. _{SARNAK, Quantum ergodicity of eigenfunctions} on

PSL2(Z)BH2.

Inst. Hautes. Etudes Sci. Publ. Math. 81 _(1995), 207-237.

[10] Z. _LUO, Z. _RUDNICK, P. _SARNAK, On _{Selberg’s eigenvalue Conjecture.} Geom. Funct. Anal.

5 _(1995), 387-401.

[11] R. A. _RANKIN, Contribution to the theory of Ramanujan’s function 03C4(n) and similar arith-metical functions. Proc. _Cambridge. Phil. Soc. 35 _(1939), 3512014372.

[12] P. SARNAK, Class number _{of infinitely binary quadratic forms.} J. Number Theory 15 _(1982), 229-247.

[13] A. B. _{VENKOV, Spectral theory of automorphic functions, (in Russian). Trudy} Math. Inst. Steklova 153 _(1981), 1-171.

Yingchun CAI

Department of Mathematics

Shanghai University