Statistics for low-lying zeros of symmetric power L-functions in the level aspect

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L-functions in the level aspect

Guillaume Ricotta, Emmanuel Royer

To cite this version:

Guillaume Ricotta, Emmanuel Royer. Statistics for low-lying zeros of symmetric power L-functions

in the level aspect. Forum Mathematicum, De Gruyter, 2011, 23 (5), pp.969-1028. �hal-00138457�

hal-00138457, version 1 - 26 Mar 2007

L

GUILLAUMERICOTTAANDEMMANUELROYER

Abstra t. We study one-level and two-level densities for low lying zerosofsymmetri power

L

-fun tionsinthelevelaspe t. Itallowsusto ompletelydeterminethesymmetrytypesofsomefamiliesofsymmetri power

L

-fun tionswithpres ribedsignoffun tionalequation. Wealso omputethe momentsofone-level densityand exhibitmo k-Gaussian behaviordis overedbyHughes&Rudni k.

Contents

1. Introdu tion andstatement of the results 2 1.1. Des ription ofthe familiesof

L

-fun tions studied 2 1.2. Symmetry type of thesefamilies 3 1.2.1. (Signed) asymptoti expe tationof theone-level density 4 1.2.2. Sket h oftheproof 5 1.2.3. (Signed) asymptoti expe tationof thetwo-leveldensity 7 1.3. Asymptoti moments of theone-leveldensity 8 1.4. Organisation ofthepaper 10 2. Automorphi and probabilisti ba kground 10 2.1. Automorphi ba kground 10 2.1.1. Overviewof holomorphi uspforms 10 2.1.2. Chebyshev polynomials and He ke eigenvalues 13 2.1.3. Overviewof

L

-fun tions asso iated to primitive uspforms 13 2.1.4. Overviewof symmetri power

L

-fun tions 14 2.2. Probabilisti ba kground 15 3. Mainte hni al ingredientsof this work 17 3.1. Largesieveinequalities for Kloostermansums 17 3.2. Riemann's expli itformulafor symmetri power

L

-fun tions 19 3.3. Contribution of theoldforms 21 4. Linear statisti sfor low-lying zeros 23 4.1. Densityresults for familiesof

L

-fun tions 23 4.2. Asymptoti expe tationof the one-level density 24 4.3. Signedasymptoti expe tationofthe one-leveldensity 27 5. Quadrati statisti sfor low-lying zeros 29 5.1. Asymptoti expe tationofthetwo-leveldensityandasymptoti

varian e 29

5.2. Signed asymptoti expe tation of the two-level density and

signed asymptoti varian e 31 6. First asymptoti momentsofthe one-leveldensity 33 Date:VersionofMar h26, 2007.

6.1. Onesome useful ombinatorial identity 35 6.2. Proof ofthe rst bulletofproposition 6.1 35 6.3. Proof ofthethird bulletofproposition 6.1 37 6.4. Proof ofthe se ond bulletofproposition 6.1 40 Appendix A. Analyti and arithmeti toolbox 44 A.1. Onsmooth dyadi partitionsof unity 44 A.2. OnBessel fun tions 45 A.3. Basi fa ts onKloosterman sums 46

Referen es 46

A knowledgementsThisworkbeganwhilebothauthorssharedthehospitality of Centre de Re her hes Mathématiques (Montréal) during the theme year "Analysis in Number Theory" (rst semester of 2006). We would like to thank C. David, H. Darmon and A. Granville for their invitation. This work was essentially ompleted inseptember 2006 in CIRM(Luminy)atthe o asion of J.-M. Deshouillers'sixtieth birthday. We would like towish him the best.

1. Introdu tion and statement of the results

1.1. Des ription of the families of

L

-fun tions studied. The purpose of this paper is to ompute various statisti s asso iated to low-lying zeros of severalfamilies ofsymmetri power

L

-fun tions inthelevel aspe t. First of all, we give a short des ription of these families. To any primitive holo-morphi usp form

f

of prime level

q

and even weight

1

κ > 2

(see Ÿ 2.1 for the automorphi ba kground) say

f ∈ H

∗

κ

(q)

,one an asso iate its

r

-th symmetri power

L

-fun tiondenotedby

L(Sym

r

_{f, s)}

for anyinteger

r > 1

. Itisgivenbyanexpli itabsolutely onvergentEulerprodu t ofdegree

r + 1

on

ℜe s > 1

(seeŸ 2.1.4). The ompleted

L

-fun tionisdened by

Λ(Sym

r

f, s) := (q

r

)

s/2

L

_∞

(Sym

r

f, s)L(Sym

r

f, s)

where

L

∞

(Sym

r

f, s)

isaprodu tof

r+1

expli it

Γ

R

-fa tors(seeŸ2.1.4)and

q

r

is the arithmeti ondu tor. We will need some ontrol on the analyti behaviourofthis fun tion. Unfortunately,su hinformation isnot urrently known in all generality. We sum up our main assumption in the following statement.

Hypothesis

Nice(r, f )

 Thefun tion

Λ (Sym

r

_{f, s)}

isa ompleted

L

-fun tion in the sense that it satises the following ni e analyti properties:

•

it an be extended toan holomorphi fun tion of order

1

on

C

,

•

it satises a fun tionalequation of the shape

Λ(Sym

r

f, s) = ε (Sym

r

f ) Λ(Sym

r

_{f, 1 − s)}

1

Inthispaper,theweight

κ

isaxed evenintegerandthelevel

q

goestoinnityamong theprimenumbers.

where thesign

ε (Sym

r

_{f ) = ±1}

ofthefun tionalequation isgivenby

ε (Sym

r

f ) :=

(

+1

if

r

is even

,

ε

f

(q) × ε(κ, r)

otherwise (1.1) with

ε(κ, r) := i(

r+1

2

)

2

(κ−1)+

r+1

2

=



















i

κ

if

r ≡ 1 (mod 8)

,

−1

if

r ≡ 3 (mod 8),

−i

κ

if

r ≡ 5 (mod 8),

+1

if

r ≡ 7 (mod 8)

and

ε

f

(q) = ±1

is dened in (2.15) and onlydepends on

f

and

q

. Remark1 Hypothesis

Nice(r, f )

isknownfor

r = 1

(E.He ke[10 12℄),

r = 2

thanks to the work of S. Gelbart and H.Ja quet [8℄ and

r = 3, 4

from the worksofH.Kimand F. Shahidi[20 22℄.

Weaimat studyingthe low-lyingzerosforthefamilyof

L

-fun tionsgiven by

F

r

:=

[

q

prime

{L(Sym

r

f, s), f ∈ H

κ

∗

(q)}

for any integer

r > 1

. Note that when

r

is even, the sign of thefun tional equation ofany

L(Sym

r

_{f, s)}

is onstant ofvalue

+1

but when

r

isodd,this isdenitely notthe ase. Asa onsequen e,itisverynaturalto understand the low-lyingzeros for the subfamilies given by

F

r

ε

:=

[

q

prime

{L(Sym

r

f, s), f ∈ H

κ

∗

(q), ε (Sym

r

f ) = ε}

for anyoddinteger

r > 1

and for

ε = ±1

.

1.2. Symmetry type of these families. Oneof thepurposeofthis work istodetermine thesymmetrytypeofthefamilies

F

r

and

F

ε

r

for

ε = ±1

and for anyinteger

r > 1

(seeŸ4.1fortheba kgroundonsymmetrytypes). The following theoremis aqui ksummaryof the symmetrytypesobtained. Theorem ALet

r > 1

be any integer and

ε = ±1

. We assume that hypoth-esis

Nice(r, f )

holds forany prime number

q

andany primitiveholomorphi usp form of level

q

andeven weight

κ > 2

. The symmetry group

G(F

r

)

of

F

r

isgiven by

G(F

r

) =

(

Sp

if

r

iseven,

O

otherwise. If

r

is oddthen the symmetrygroup

G(F

ε

r

)

of

F

ε

r

isgiven by

G(F

r

ε

) =

(

SO(even)

if

ε = +1

,

SO(odd)

otherwise.

Remark 2 Itfollowsinparti ularfromthevalueof

ε (Sym

r

_{f )}

givenin(1.1) that, if

r

iseven,then

Sym

r

_f

hasnot thesamesymmetrytypethan

f

and, if

r

isodd,then

f

and

Sym

r

_f

have thesame symmetrytype ifandonly if

or

r ≡ 5 (mod 8)

and

κ ≡ 2 (mod 4)

or

r ≡ 7 (mod 8).

Remark3 NotethatwedonotassumeanyGeneralisedRiemannHypothesis for the symmetri power

L

-fun tions.

In order to prove theorem A,we ompute eitherthe(signed) asymptoti expe tationoftheone-leveldensityorthe(signed)asymptoti expe tationof thetwo-leveldensity. Theresultsaregiveninthenexttwose tionsinwhi h

ε = ±1

,

ν

will always be a positive real number,

Φ, Φ

1

and

Φ

2

will always standforevenS hwartzfun tionswhoseFouriertransforms

Φ, c

b

Φ

1

and

Φ

c

2

are ompa tly supportedin

[−ν, +ν]

and

f

will alwaysbe aprimitive holomor-phi uspform ofprimelevel

q

andevenweight

κ > 2

for whi h hypothesis

Nice(r, f )

holds. We referto Ÿ 2.2for theprobabilisti ba kground.

1.2.1. (Signed)asymptoti expe tationoftheone-leveldensity. Theone-level density (relatively to

Φ

) of

Sym

r

_f

is dened by

D

1,q

[Φ; r](f ) :=

X

ρ, Λ(Sym

r

_f,ρ)=0

Φ



log (q

r

)

2iπ



ℜe ρ −

1

₂

+ i ℑm ρ



where the sumisoverthe non-trivialzeros

ρ

of

L(Sym

r

_{f, s)}

with multipli -ities. The asymptoti expe tation of the one-level density isbydenition

lim

q

prime

q→+∞

[r]

X

f ∈H

∗

κ

(q)

ω

q

(f )D

1,q

[Φ; r](f )

where

ω

q

(f )

istheharmoni weight denedin(2.7)and similarlythesigned asymptoti expe tation ofthe one-leveldensity isbydenition

lim

q

prime

q→+∞

2[r]

X

f ∈H

∗

κ

(q)

ε(Sym

r

_{f )=ε}

ω

q

(f )D

1,q

[Φ; r](f )

when

r

is odd.

Theorem BLet

r > 1

be any integer and

ε = ±1

. We assumethat hypoth-esis

Nice(r, f )

holds forany prime number

q

andany primitiveholomorphi usp form oflevel

q

andeven weight

κ > 2

andalso that

θ

is admissible(see hypothesis

H

2

(θ)

page 17). Let

ν

1,max

(r, κ, θ) :=



1 −

1

2(κ − 2θ)



2

r

2

.

If

ν < ν

1,max

(r, κ, θ)

then the asymptoti expe tation of the one-level density is

b

Φ(0) +

(−1)

r+1

2

Φ(0).

Let

ν

_1,max

ε

(r, κ, θ) := inf



ν

1,max

(r, κ, θ),

3

r(r + 2)



.

If

r

is odd and

ν < ν

ε

1,max

(r, κ, θ)

then the signed asymptoti expe tation of

the one-level densityis

b

Φ(0) +

(−1)

r+1

2

Φ(0).

Remark 4 The rst part of Theorem B reveals that the symmetry type of

F

r

is

G(F

r

) =











Sp

if

r

iseven,

O

if

r = 1

,

SO(even)

or

O

or

SO(odd)

if

r > 3

isodd.

We annot de ide between the three orthogonal groups when

r > 3

is odd sin e in this ase

ν

1,max

(r, κ, θ) < 1

but the omputation of the two-level densities will enableus to de ide. Notealso thatwe go beyond thesupport

[−1, 1]

when

r = 1

asIwanie ,Luo&Sarnak[18℄(Theorem1.1)butwithout doinganysubtlearithmeti analysisofKloostermansums. Also,A.Güloglu in [9, Theorem 1.2℄ established some density result for the same family of

L

-fun tions but when the weight

κ

goes to innity and the level

q

is xed. It turnsout thatwere overthesame onstrainton

ν

when

r

isevenbut we getabetter resultwhen

r

isodd. This an beexplainedbythefa tthatthe analyti ondu tor of any

L(Sym

r

_{f, s)}

with

f

in

H

∗

κ

(q)

whi his ofsize

q

r

_×

(

κ

r

if

r

iseven

κ

r+1

otherwise

is slightly largerinhis ase than inourswhen

r

isodd.

Remark 5These ondpartofTheoremBrevealsthatif

r

isoddand

ε = ±1

then the symmetrytypeof

F

ε

r

is

G(F

r

ε

) = SO(even)

or

O

or

SO(odd).

Here

ν

isalwaysstri tlysmallerthanone andwearenotableto re over the result of [18, Theorem 1.1℄ without doing some arithmeti on Kloosterman sums.

1.2.2. Sket hoftheproof. Wegivehereasket hoftheproofoftherstpart of Theorem B namely we briey explain how to determine the asymptoti expe tationoftheone-leveldensityassumingthathypothesis

Nice(r, f )

holds for any prime number

q

and any primitive holomorphi usp form of level

q

and even weight

κ > 2

and also that

θ

is admissible. The rst step onsistsintransformingthesumoverthezerosof

Λ(Sym

r

_{f, s)}

whi ho urs in

D

1,q

[Φ; r](f )

into a sum over primes. This is done via some Riemann's expli it formula for symmetri power

L

-fun tions stated inProposition 3.8 whi h leads to

D

1,q

[Φ; r](f ) = b

Φ(0)+

(−1)

r+1

2

Φ(0)+P

1

q

[Φ; r](f )+

r−1

X

m=0

(−1)

m

P

_q

2

[Φ; r, m](f )+o(1)

where

P

_q

1

_{[Φ; r](f ) := −}

2

log (q

r

₎

X

p∈P

p∤q

λ

f

(p

r

)

log p

√

_p

Φ

b



log p

log (q

r

₎



.

(1.2)

Theterms

P

2

q

[Φ; r, m](f )

arealsosumsoverprimeswhi hlooklike

P

1

q

[Φ; r](f )

but anbeforgotteninrstapproximationsin ethey anbethoughtassums over squares of primes whi h are easier to deal with. The se ond step on-sists inaveraging overall the

f

in

H

∗

κ

(q)

. While doing this,theasymptoti expe tationof theone-level density

b

Φ(0) +

(−1)

r+1

2

Φ(0)

naturally appears andwe need to show that

−

_{log (q}

2

_r

₎

X

p∈P

p∤q





X

f ∈H

∗

κ

(q)

ω

q

(f )λ

f

(p

r

)





log p

_√

p

Φ

b



log p

log (q

r

₎



is a remainder term provided that the support

ν

of

Φ

is small enough. We applysomesuitabletra eformulagiveninProposition2.2inordertoexpress the previous average of He ke eigenvalues. We annot dire tlyapply Peter-son's tra e formula sin e there may be some old forms of level

q

espe ially when the weight

κ

is large. Nevertheless, these oldforms areautomati ally of level

1

sin e

q

isprime and their ontribution remains negligible. So, we have to bound

−

4πi

κ

log (q

r

₎

X

p∈P

p∤q

X

c>1

q|c

S(1, p

r

; c)

c

J

κ−1



4π

√

p

r

c



log p

√

_p

Φ

b



log p

log (q

r

₎



where

S(1, p

r

_{; c)}

is aKloosterman sumand whi h an bewritten as

−

4πi

κ

log (q

r

₎

X

c>1

q|c

X

m>1

a

m

S(1, m; c)

c

g(m; c)

where

a

m

:= 1

_[1,q

r2ν

_]

(m)

log m

rm

1/(2r)

×

(

1

if

m = p

r

for some prime

p 6= q,

0

otherwise and

g(m; c) := J

_κ−1



4π

√

m

c



b

Φ



log m

r log (q

r

₎



.

We apply the large sieve inequality for Kloosterman sums given in proposi-tion 3.4. Itentailsthat if

ν 6 2/r

2

thensu h quantityisbounded by

≪

ε

q(

κ−1

2

−θ

)

(r

2

ν−2)+ε

+ q(

κ

2

−θ

)

r

2

ν−

(

κ−

1

2

−2θ

)

+ε

.

This is an admissible error term if

ν < ν

1,max

(r, κ, θ)

. We fo us onthe fa t that we didanyarithmeti analysisofKloosterman sums to getthis result. Of ourse, the power of spe tral theory of automorphi forms is hidden in the largesieveinequalities for Kloosterman sums.

1.2.3. (Signed)asymptoti expe tationofthetwo-leveldensity. Thetwo-level density of

Sym

r

_f

(relatively to

Φ

1

and

Φ

2

) isdened by

D

2,q

[Φ

1

, Φ

2

; r](f ) :=

X

(j

1

,j

2

)∈E(f,r)

2

j

1

6=±j

2

Φ

1



b

ρ

(j

1

)

f,r



Φ

2



b

ρ

(j

2

)

f,r



.

For more pre ision on the numbering of the zeros, we refer to Ÿ 3.2. The asymptoti expe tation ofthetwo-leveldensityis bydenition

lim

q

prime

q→+∞

[r]

X

f ∈H

∗

κ

(q)

ω

q

(f )D

2,q

[Φ

1

, Φ

2

; r](f )

andsimilarlythesigned asymptoti expe tation ofthetwo-leveldensityisby denition

lim

q

prime

q→+∞

2[r]

X

f ∈H

∗

κ

(q)

ε(Sym

r

f )=ε

ω

q

(f )D

2,q

[Φ

1

, Φ

2

; r](f )

when

r

is odd and

ε = ±1

.

Theorem CLet

r > 1

be any integerand

ε = ±1

. We assume that hypoth-esis

Nice(r, f )

holds forany prime number

q

andany primitiveholomorphi usp form of level

q

andeven weight

κ > 2

. If

ν < 1/r

2

then the asymptoti expe tation of the two-level density is



c

Φ

1

(0) +

(−1)

r+1

2

Φ

1

(0)

 

c

Φ

2

(0) +

(−1)

r+1

2

Φ

2

(0)



+ 2

Z

R

|u|c

Φ

1

(u)c

Φ

2

(u) du − 2 [

Φ

1

Φ

2

(0) +



(−1)

r

+

1

2N+1

(r)

2



Φ

1

(0)Φ

2

(0).

If

r

is odd and

ν < 1/(2r(r + 2))

then the signed asymptoti expe tation of the two-level density is



c

Φ

1

(0) +

1

2

Φ

1

(0)

 

c

Φ

2

(0) +

1

2

Φ

2

(0)



+ 2

Z

R

|u|c

Φ

1

(u)c

Φ

2

(u) du − 2 [

Φ

1

Φ

2

(0) − Φ

1

(0)Φ

2

(0)

+ 1

_{−1}

(ε)Φ

1

(0)Φ

2

(0).

Remark 6 We have justseen thatthe omputation ofthe one-level density already reveals that the symmetry type of

F

r

is

Sp

when

r

is even. The asymptoti expe tation of thetwo-level density also oin ides with theone of

Sp

(see [19, Theorem A.D.2.2℄ or [26, Theorem 3.3℄). When

r > 3

is odd, the rst part of Theorem C together with a result of Katz & Sarnak (see [19, Theorem A.D.2.2℄ or [26, Theorem 3.2℄) imply thatthe symmetry type of

F

r

is

O

.

Remark 7The se ond part of Theorem C and a result of Katz & Sarnak (see [19, Theorem A.D.2.2℄ or [26, Theorem 3.2℄) imply thatthe symmetry type of

F

ε

In order to prove TheoremC, we need to determine theasymptoti vari-an e ofthe one-leveldensitywhi h isdened by

lim

q

prime

q→+∞

[r]

X

f ∈H

∗

κ

(q)

ω

q

(f )



D

1,q

[Φ; r](f ) −

X

g∈H

∗

κ

(q)

ω

q

(g)D

1,q

[Φ; r](g)





2

andthesignedasymptoti varian e oftheone-leveldensitywhi hissimilarly dened by

lim

q

prime

q→+∞

2[r]

X

f ∈H

∗

κ

(q)

ε(Sym

r

_{f )=ε}

ω

q

(f )







D

1,q

[Φ; r](f ) − 2[r]

X

g∈H

∗

κ

(q)

ε(Sym

r

_g)=ε

ω

q

(g)D

1,q

[Φ; r](g)









2

when

r

is odd and

ε = ±1

.

TheoremD Let

r > 1

be any integerand

ε = ±1

. We assumethat hypoth-esis

Nice(r, f )

holds forany prime number

q

andany primitiveholomorphi usp form of level

q

andeven weight

κ > 2

. If

ν < 1/r

2

then the asymptoti varian e of the one-level densityis

2

Z

R

|u|b

Φ

2

(u) du.

If

r

isoddand

ν < 1/(2r(r + 2))

then the signed asymptoti varian e of the one-level densityis

2

Z

R

|u|b

Φ

2

(u) du.

1.3. Asymptoti momentsoftheone-leveldensity. Lastbutnotleast, we ompute the asymptoti

m

-th moment of the one-level density whi h is dened by

lim

q

prime

q→+∞

[r]

X

f ∈H

∗

κ

(q)

ω

q

(f )



D

1,q

[Φ; r](f ) −

X

g∈H

∗

κ

(q)

ω

q

(g)D

1,q

[Φ; r](g)





m

for anyinteger

m > 1

.

Theorem ELet

r > 1

be any integer and

ε = ±1

. We assume that hypoth-esis

Nice(r, f )

holds forany prime number

q

andany primitiveholomorphi usp form of level

q

and even weight

κ > 2

. If

mν < 4 /(r(r + 2))

then the asymptoti

m

-th momentof the one-level densityis

(

0

if

m

is odd,

2

R

_R

_|u|b

Φ

2

_{(u) du ×}

m!

2

m/2

₍

m

2

)

!

otherwise.

Remark 8 Thisresultisanothereviden eformo k-Gaussianbehaviour(see [13 15℄ forinstan e).

Remark 9 We omputetherstasymptoti momentsoftheone-leveldensity. These omputations allow to ompute the asymptoti expe tation of the rst level-densities[13, Ÿ1.2℄. We willusethespe i aseoftheasymptoti expe tationof thetwo-leveldensity andtheasymptoti varian e inŸ5.1.

Letussket htheproofofTheoremEbyexplainingtheoriginofthemain term. We have to evaluate

X

06ℓ6m

06α6ℓ



m

ℓ



ℓ

α



R(q)

ℓ−α

E

h

q



P

_q

1

[Φ; r]

m−ℓ

P

_q

2

[Φ; r]

α



(1.3) where

P

1

q

[Φ; r]

hasbeen dened in(1.2) ,

P

_q

2

_{[Φ; r](f ) = −}

2

log(q

r

₎

r

X

j=1

(−1)

r−j

X

p∈P

p∤q

λ

f

p

2j

log p

p

Φ

b



2 log p

log(q

r

₎



and

R(q)

satises

R(q) = O



1

log q



.

The mainterm omes fromthe ontribution

ℓ = 0

inthesum(1.3) . Usinga ombinatorial lemma, we rewritethis main ontribution as

(−2)

m

log

m

(q

r

₎

m

X

s=1

X

σ∈P (m,s)

X

i

1

,...,i

s

distin t

E

h

q

s

Y

u=1

λ

f

p

b

r

i

u

̟

(σ)

u

!

where

P (m, s)

isthe set ofsurje tive fun tions

σ : {1, . . . , α} ։ {1, . . . , s}

su h thatfor any

j ∈ {1, . . . , s}

, either

σ(j) = 1

or there exists

k < j

su h that

σ(j) = σ(k) + 1

and for any

j ∈ {1, . . . , s}

̟

(σ)

_j

:= #σ

−1

_({j}).

(b

p

i

)

i>1

standsforthein reasing sequen eofprimenumbersdierentfrom

q

.

Linearising ea h

λ

f

p

b

r

i

u

̟

u

(σ)

intermsof

λ

f



b

p

j

u

i

u



with

j

u

runsoverintegers in

[0, r̟

(σ)

u

]

and using a tra e formula to prove that the only

σ ∈ P (m, s)

leading to a prin ipal ontribution satisfy

̟

(σ)

j

= 2

for any

j ∈ {1, . . . , s}

, wehave to estimate

(−2)

m

log

m

(q

r

₎

m

X

s=1

X

σ∈P (m,s)

∀j∈{1,...,s},̟

(σ)

j

=2

X

i

1

,...,i

s

distin t

s

Y

u=1

log

2

_(b

p

i

u

)

b

p

i

u

b

Φ

2



log b

p

i

u

log (q

r

₎



.

(1.4)

This sumvanishesif

m

isoddsin e

s

X

j=1

̟

(σ)

_j

= m

and itremains to prove the formula for

m

even. In this ase, and sin e we already omputed the moment for

m = 2

, we dedu e from (1.4) that the main ontribution is

E

h

q

(P

q

1

[Φ; r]

2

) × #

n

σ ∈ P (m, m/2): ̟

(σ)

j

= 2 (∀j)

o

and we on lude by omputing

#

n

_{σ ∈ P (m, m/2): ̟}

_j

(σ)

_{= 2 (∀j)}

o

=

m!

2

m/2

m

2

!

.

Proving that the other terms lead to error terms is done by implementing similarideas,butrequiresespe iallyforthedoubleprodu ts(namelyterms implying both

P

1

q

and

P

2

q

) mu h more ombinatorial te hni alities.

1.4. Organisation of the paper. Se tion2 ontains theautomorphi and probabilisti ba kground whi h is needed to be able to read this paper. In parti ular, we give here the a urate denition of symmetri power

L

-fun tions and the properties of Chebyshev polynomials useful in se tion 6. In se tion3,wedes ribethe mainte hni al ingredientsofthis worknamely large sieve inequalities for Kloosterman sums and Riemann's expli it for-mula for symmetri power

L

-fun tions. In se tion 4, some standard fa ts about symmetry groups are given and the omputation of the (signed) as-ymptoti expe tation of the one-level density is done. The omputations of the (signed) asymptoti expe tation, ovarian e and varian e ofthe two-leveldensityaredoneinse tion5whereasthe omputationoftheasymptoti moments ofthe one-leveldensityis provided inse tion6. Somewell-known fa ts about Kloosterman sums arere alled inappendix A.

Notation We write

P

for the set of primenumbers andthe mainparameter in this paper is a prime number

q

, whose name is the level, whi h goes to innity among

P

. Thus, if

f

and

g

are some

C

-valued fun tions of the real variable then the notations

f (q) ≪

A

g(q)

or

f (q) = O

A

(g(q))

mean that

|f(q)|

is smaller than a " onstant" whi h only depends on

A

times

g(q)

at least for

q

a large enough prime number and similarly,

f (q) = o(1)

means that

f (q) → 0

as

q

goestoinnityamongtheprimenumbers. We willdenote by

ε

an absolute positive onstant whose denition may vary from one line to the next one. The hara teristi fun tionof a set

S

willbe denoted

1

S

.

2. Automorphi and probabilisti ba kground 2.1. Automorphi ba kground.

2.1.1. Overviewofholomorphi uspforms. Inthisse tion,were allgeneral fa ts about holomorphi uspforms. A referen eis[16℄.

Generalities We write

Γ

0

(q)

for the ongruen e subgroupoflevel

q

whi h a ts on the upper-half plane

H

. A holomorphi fun tion

f : H 7→ C

whi h satises

∀



a b

c d



∈ Γ

0

(q), ∀z ∈ H,

f



az + b

cz + d



= (cz + d)

κ

f (z)

and vanishes at the usps of

Γ

0

(q)

is a holomorphi usp form of level

q

, even weight

κ > 2

. We denote by

S

κ

(q)

this spa e of holomorphi usp forms whi h isequipped withthePeterson innerprodu t

hf

1

, f

2

i

q

:=

Z

Γ

0

(q)\H

y

κ

f

1

(z)f

2

(z)

dx dy

y

2

.

The Fourier expansion at the usp

∞

of anysu h holomorphi uspform

f

is givenby

∀z ∈ H,

f (z) =

X

n>1

ψ

f

(n)n

(κ−1)/2

e(nz)

where

e(z) := exp (2iπz)

for any omplex number

z

. The He ke operators a t on

S

κ

(q)

by

T

ℓ

(f )(z) :=

1

√

ℓ

X

ad=ℓ

(a,q)=1

X

06b<d

f



az + b

d



for any

z ∈ H

. If

f

isan eigenve tor of

T

ℓ

,wewrite

λ

f

(ℓ)

the orresponding eigenvalue. We anprovethat

T

ℓ

ishermitian if

ℓ > 1

isanyinteger oprime with

q

and that

T

ℓ

1

◦ T

ℓ

2

=

X

d|(ℓ

1

,ℓ

2

)

(d,q)=1

T

_ℓ

₁

_ℓ

₂

_/d

2

(2.1)

for anyintegers

ℓ

1

, ℓ

2

>

1

. By Atkin &Lehner theory[1℄,we geta splitting of

S

κ

(q)

into

S

o

κ

(q) ⊕

⊥

h·,·iq

S

κ

n

(q)

where

S

_κ

o

(q) := Vect

C

{f(qz), f ∈ S

κ

(1)} ∪ S

κ

(1),

S

_κ

n

(q) := (S

_κ

o

(q))

⊥

h·,·iq

where "o" stands for "old" and "n" for "new". Note that

S

o

κ

(q) = {0}

if

κ < 12

or

κ = 14

. Thesetwo spa es are

T

ℓ

-invariant for any integer

ℓ > 1

oprime with

q

. A primitive uspform

f ∈ S

n

κ

(q)

isan eigenfun tion ofany operator

T

ℓ

for any integer

ℓ > 1

oprime with

q

whi h is new and arith-meti ally normalisednamely

ψ

f

(1) = 1

. Su han element

f

isautomati ally aneigenfun tion oftheotherHe keoperatorsandsatises

ψ

f

(ℓ) = λ

f

(ℓ)

for anyinteger

ℓ > 1

. Moreover, if

p

isa prime number, dene

α

f

(p)

,

β

f

(p)

as the omplexroots ofthe quadrati equation

X

2

_{− λ}

f

(p)X + ε

q

(p) = 0

(2.2)

where

ε

q

denotesthetrivialDiri hlet hara terofmodulus

q

. Thenitfollows from the work ofEi hler, Shimura, IgusaandDeligne that

|α

f

(p)|, |β

f

(p)| 6 1

for anyprime number

p

and so

∀ℓ > 1,

|λ

f

(ℓ)| 6 τ(ℓ).

(2.3)

The set of primitive usp forms is denoted by

H

∗

κ

(q)

. It is an orthogonal basis of

S

n

κ

(q)

. Let

f

be a holomorphi usp form with He ke eigenvalues

(λ

f

(ℓ))

_(ℓ,q)=1

. The omposition property (2.1) entails that for any integer

relations hold:

ψ

f

(ℓ

1

)λ

f

(ℓ

2

) =

X

d|(ℓ

1

,ℓ

2

)

(d,q)=1

ψ

f

ℓ

1

ℓ

2



d

2

,

(2.4)

ψ

f

(ℓ

1

ℓ

2

) =

X

d|(ℓ

1

,ℓ

2

)

(d,q)=1

µ(d)ψ

f

(ℓ

1

/d) λ

f

(ℓ

2

/d)

(2.5)

and these relations hold for any integers

ℓ

1

, ℓ

2

>

1

if

f

is primitive. The adjointness relationis

λ

f

(ℓ) = λ

f

(ℓ),

ψ

f

(ℓ) = ψ

f

(ℓ)

(2.6)

for any integer

ℓ > 1

oprime with

q

and this remains true for any integer

ℓ > 1

if

f

isprimitive.

Tra eformulas Weneedtwodenitions. Theharmoni weightasso iated to any

f

in

S

κ

(q)

is dened by

ω

q

(f ) :=

Γ(κ

− 1)

(4π)

κ−1

_{hf, fi}

_q

.

(2.7) For anynatural integer

m

and

n

,the

∆

q

-symbolisgiven by

∆

q

(m, n) := δ

m,n

+ 2πi

κ

X

c>1

q|c

S(m, n; c)

c

J

κ−1



4π

√

mn

c



(2.8)

where

S(m, n; c)

isa Kloosterman sumdened inappendixA.3and

J

κ−1

is

a Bessel fun tionof rstkind dened inappendixA.2.The following propo-sition isPeterson'stra e formula.

Proposition 2.1 If

H

κ

(q)

isany orthogonal basis of

S

κ

(q)

then

X

f ∈H

κ

(q)

ω

q

(f )ψ

f

(m)ψ

f

(n) = ∆

q

(m, n)

(2.9)

for any integers

m

and

n

.

H. Iwanie , W. Luo & P.Sarnak proved in [18℄ a useful variation of Pe-terson's tra e formula whi h is an average over only primitive usp forms. Thisismore onvenient whentherearesomeoldformswhi histhe asefor instan ewhentheweight

κ

islarge. Let

ν

bethearithmeti fun tiondened by

ν(n) := n

Y

p|n

(1 + 1/p)

for anyinteger

n > 1

.

Proposition 2.2 (H. Iwanie ,W. Luo & P.Sarnak(2001)) If

n, q

2

_{| q}

and

q ∤ m

then

X

f ∈H

∗

κ

(q)

ω

q

(f )λ

f

(m)λ

f

(n) = ∆

q

(m, n) −

1

qν((n, q))

X

ℓ|q

∞

1

ℓ

∆

1

mℓ

2

_{, n}

_.

(2.10)

Remark 2.3The rst term in (2.10) is exa tly the term whi h appears in (2.9) whereas the se ond term in(2.10) will be usually verysmall asan old form omes from a form oflevel

1

! Thus, everything worksinpra ti e asif there wereno oldforms in

S

κ

(q)

.

2.1.2. Chebyshev polynomials and He ke eigenvalues. Let

p 6= q

a prime number and

f ∈ H

∗

κ

(q)

. The multipli ativity relation(2.4) leads to

X

r>0

λ

_f

(p

r

)t

r

=

1

1 − λ

f

(p)t + t

2

.

It follows that

λ

f

(p

r

) = X

r

(λ

f

(p))

(2.11) where the polynomials

X

r

aredened by theirgenerating series

X

r>0

X

r

(x)t

r

=

1

1 − xt + t

2

.

Theyare alsodened by

X

r

(2 cos θ) =

sin ((r + 1)θ)

sin (θ)

.

ThesepolynomialsareknownastheChebyshevpolynomials ofse ondkind. Ea h

X

r

has degree

r

, is even if

r

is even and odd otherwise. The family

{X

r

}

r>0

isa basisfor

Q[X]

,orthonormal withrespe tto theinnerprodu t

hP, Qi

ST

:=

1

π

Z

2

−2

P (x)Q(x)

r

1 −

x

2

4

dx.

In parti ular,for anyinteger

̟ > 0

we have

X

_r

̟

=

r̟

X

j=0

x(̟, r, j)X

j

(2.12) with

x(̟, r, j) := hX

r

̟

, X

j

i

ST

=

2

π

Z

π

0

sin

̟

((r + 1)θ) sin ((j + 1)θ)

sin

̟−1

_(θ)

dθ.

(2.13) The following relations areusefulinthis paper

x(̟, r, j) =











1

if

j = 0

and

̟

iseven,

0

if

j

isoddand

r

iseven,

0

if

j = 0

,

̟ = 1

and

r > 1

.

(2.14)

2.1.3. Overviewof

L

-fun tionsasso iated toprimitive usp forms. Let

f

in

H

_κ

∗

(q)

. We dene

L(f, s) :=

X

n>1

λ

f

(n)

n

s

=

Y

p∈P



1 −

α

f

_p

(p)

_s



₋₁



1 −

β

f

_p

(p)

_s



₋₁

whi h is an absolutely onvergent and non-vanishing Diri hlet series and Euler produ ton

ℜe s > 1

and also

where

Γ

R

(s) := π

−s/2

_{Γ (s/2)}

asusual. Thefun tion

Λ(f, s) := q

s/2

L

_∞

(f, s)L(f, s)

is a ompleted

L

-fun tion in the sense that it satises the following ni e analyti properties:

•

the fun tion

Λ(f, s)

an be extended to an holomorphi fun tion of order

1

on

C

,

•

the fun tion

Λ(f, s)

satisesafun tional equationof theshape

Λ(f, s) = i

κ

ε

f

(q)Λ(f, 1 − s)

where

ε

f

(q) = −

√

qλ

f

(q) = ±1.

(2.15)

2.1.4. Overview of symmetri power

L

-fun tions. Let

f

in

H

∗

κ

(q)

. For any

natural integer

r > 1

,thesymmetri

r

-th power asso iated to

f

is given by the following Euler produ tofdegree

r + 1

L(Sym

r

f, s) :=

Y

p∈P

L

p

(Sym

r

f, s)

where

L

p

(Sym

r

f, s) :=

r

Y

i=0



1 −

α

f

(p)

i

_β

f

(p)

r−i

p

s



−1

for any prime number

p

. Let us remark that the lo alfa tors of this Euler produ tmaybe written as

L

p

(Sym

r

f, s) =

r

Y

i=0



1 −

α

f

(p)

_p

_s

2i−r



−1

for anyprime number

p 6= q

and

L

q

(Sym

r

f, s) = 1 −

λ

f

(q)

r

q

s

= 1 −

λ

f

(q

r

)

q

s

as

α

f

(p) + β

f

(p) = λ

f

(p)

and

α

f

(p)β

f

(p) = ε

q

(p)

for any prime number

p

a ording to(2.2) . On

ℜe s > 1

,this Eulerprodu tisabsolutely onvergent and non-vanishing. Wealsodenes[4,(3.16) and(3.17)℄alo alfa tor at

∞

whi h isgiven bya produ tof

r + 1

Gamma fa torsnamely

L

_∞

(Sym

r

f, s) :=

Y

06a6(r−1)/2

Γ

R

(s + (2a + 1)(κ − 1)/2) Γ

R

(s + 1 + (2a + 1)(κ − 1)/2)

if

r

isoddand

L

_∞

(Sym

r

f, s) := Γ

R

(s+µ

κ,r

)

Y

16a6r/2

Γ

R

(s + a(κ − 1)) Γ

R

(s + 1 + a(κ − 1))

if

r

iseven where

µ

κ,r

:=

(

1

if

r(κ − 1)/2

isodd,

0

otherwise.

All the lo al data appearing in these lo al fa tors are en apsulated in the following ompleted

L

-fun tion

Here,

q

r

is alled the arithmeti ondu tor of

Λ(Sym

r

_{f, s)}

and somehow measures the size of this fun tion. We will need some ontrol on the an-alyti behaviour of this fun tion. Unfortunately, su h information is not urrently knowninallgenerality. Ourmainassumptionisgivenin hypothe-sis

Nice(r, f )

page2. Indeed,mu hmoreisexpe tedtoholdasitisdis ussed in details in [4℄ namely the following assumption is strongly believed to be true and liesinthe spiritof Langlands program.

Hypothesis

Sym

r

_{(f )}

 There existsan automorphi uspidalself-dual repre-sentation, denoted by

Sym

r

_π

f

= ⊗

′

_{p∈P∪{∞}}

Sym

r

π

f,p

, of

GL

r+1

(A

Q

)

whose

lo al fa tors

L (Sym

r

_π

f,p

, s)

agree with the lo al fa tors

L

p

(Sym

r

_{f, s)}

for any

p

in

P ∪ {∞}

.

Note thatthelo al fa torsand thearithmeti ondu tor inthedenition of

Λ (Sym

r

_{f, s)}

andalso thesignof itsfun tionalequationwhi hallappear without anyexplanationssofar omefrom theexpli it omputationswhi h have been done via the lo al Langlands orresponden e by J. Cogdell and P. Mi hel in [4℄. Obviously, hypothesis

Nice(r, f )

is a weak onsequen e of hypothesis

Sym

r

_{(f )}

. For instan e, the uspidality ondition in hypothesis

Sym

r

(f )

entails the fa t that

Λ (Sym

r

_{f, s)}

is of order

1

whi h is ru ial for us to state a suitable expli it formula. As we will not exploit thepowerof automorphi theory in this paper, hypothesis

Nice(r, f )

is enough for our purpose. In addition, it may happen that hypothesis

Nice(r, f )

is known whereas hypothesis

Sym

r

_f

is not. Let us overview what has been done so far. For any

f

in

H

∗

κ

(q)

, hypothesis

Sym

r

_f

is known for

r = 1

(E. He ke),

r = 2

thanksto theworkof S.Gelbart andH.Ja quet[8℄ and

r = 3, 4

from the worksofH.Kim andF. Shahidi [2022℄.

2.2. Probabilisti ba kground. Theset

H

∗

κ

(q)

an beseenasa probabil-ityspa eif

•

the measurable sets areallits subsets,

•

the harmoni probability measure is denedby

µ

h

_q

(A) :=

X

h

f ∈A

1 :=

X

f ∈A

ω

q

(f )

for anysubset

A

of

H

∗

κ

(q)

.

Indeed, thereis aslight abuse hereaswe onlyknowthat

lim

q∈P

q→+∞

µ

h

_q

(H

_κ

∗

(q)) = 1

(2.16)

(see remark 3.12) whi h means that

µ

h

q

is an asymptoti  probability mea-sure. If

X

q

isameasurable omplex-valuedfun tionon

H

∗

κ

(q)

thenitisvery

natural to ompute its expe tation dened by

E

h

q

(X

q

) :=

X

h

f ∈H

∗

κ

(q)

X

q

(f ),

its varian e dened by

V

h

q

(X

q

) := E

h

q



X

q

− E

h

q

(X

q

)

and its

m

-th moments given by

M

h

q,m

(X

q

) := E

h

q



X

q

− E

h

q

(X

q

)



m



for any integer

m > 1

. If

X := (X

q

)

q∈P

is a sequen e of su h measurable omplex-valued fun tions then we may legitimely wonder if the asso iated omplex sequen es



E

h

q

(X

q

)



q∈P

,



V

h

q

(X

q

)



q∈P

,



M

h

q,m

(X

q

)



q∈P

onvergeas

q

goesto innityamongtheprimes. Ifyes, thefollowing general notations will be usedfor their limits

E

h

∞

(X) ,

V

h

∞

(X) ,

M

h

∞,m

(X)

for any natural integer

m

. In addition, these potential limits are alled asymptoti expe tation, asymptoti varian e and asymptoti

m

-th moments of

X

for anynatural integer

m > 1

.

For the end of this se tion, we assume that

r

is odd. We may remark thatthe signof thefun tionalequations ofany

L(Sym

r

_{f, s)}

when

q

goesto innity among theprimenumbers and

f

ranges over

H

∗

κ

(q)

is not onstant asit depends on

ε

f

(q)

. Let

H

_κ

ε

_{(q) := {f ∈ H}

_κ

∗

(q), ε(Sym

r

_{f ) = ε}}

where

ε = ±1

. If

f ∈ H

+1

κ

(q)

,then

Sym

r

_f

is said to be even whereas it is said to be odd if

f ∈ H

−1

κ

(q)

. It iswell-known that

lim

q∈P

q→+∞

µ

h

_q

_{({f ∈ H}

_k

∗

(q) : ε

f

(q) = ε}) =

1

2

.

Sin e

ε(Sym

r

_{f )}

is

ε

q

(f )

uptoasigndependingonlyon

κ

and

r

(by hypoth-esis

Nice(r, f )

), itfollows that

lim

q∈P

q→+∞

µ

h

_q

(H

_κ

ε

(q)) =

1

2

.

(2.17)

For

X

q

asprevious, we an ompute its signed expe tation dened by

E

h,ε

q

(X

q

) := 2

X

h

f ∈H

ε

κ

(q)

X

q

(f ),

its signed varian e dened by

V

h,ε

q

(X

q

) := E

h,ε

q



X

q

− E

h,ε

q

(X

q

)



2



and its signed

m

-th moments given by

M

h,ε

q,m

(X

q

) := E

h,ε

q



X

q

− E

h,ε

q

(X

q

)



m



foranynaturalinteger

m > 1

. In aseofexisten e,wewrite

E

h,ε

∞

(X)

,

V

h,ε

∞

(X)

and

M

h,ε

signed asymptoti varian e and signed asymptoti moments. The signed expe tationand the expe tation arelinked throughtheformula

E

h,ε

q

(X

q

) = 2

X

h

f ∈H

∗

κ

(q)

1 + ε × ε(Sym

r

f )

2

X

q

(f )

= E

h

_q

(X

q

) − ε × ε(κ, r)

√

q

X

h

f ∈H

∗

κ

(q)

λ

f

(q)X

q

(f ).

(2.18)

3. Main te hni al ingredients of this work

3.1. Large sieve inequalities for Kloosterman sums. Oneofthemain ingredientsinthisworkissomelargesieveinequalitiesforKloostermansums whi h have been established byJ.-M. Deshouillers & H. Iwanie in [5℄ and thenrenedbyV.Blomer,G.Har os &P.Mi helin[2℄. Theproofofthese large sieve inequalities relies on the spe tral theory of automorphi forms on

GL

2

(A

Q

)

. In parti ular, theauthors have to understand the size of the Fourier oe ients ofthese automorphi uspforms. We have alreadyseen that the size of the Fourier oe ients of holomorphi usp forms is well understood (2.3) but we only have partial results on thesize of the Fourier oe ientsofMaass uspforms whi hdonot omefromholomorphi forms. We introdu e the following hypothesis whi h measures the approximation towards the Ramanujan-Peterson-Selberg onje ture.

Hypothesis

H

2

(θ)

If

π := ⊗

′

p∈P∪{∞}

π

p

is any automorphi uspidal form

on

GL

2

(A

Q

)

withlo al He ke parameters

α

(1)

π

(p)

,

α

(2)

π

(p)

atanyprime num-ber

p

and

µ

(1)

π

(∞)

,

µ

(2)

π

(∞)

at innitythen

∀j ∈ {1, 2},

|α

(j)

π

(p)| 6 p

θ

for any primenumber

p

for whi h

π

p

isunramied and

∀j ∈ {1, 2},

|ℜe



µ

(j)

_π

_(∞)



_{| 6 θ}

provided

π

∞

is unramied.

Denition 3.1 We say that

θ

is admissible if

H

2

(θ)

issatised. Remark 3.2The smallest admissible value of

θ

is urrently

θ

0

=

7

64

thanks to the worksofH.Kim, F.Shahidiand P.Sarnak [20,21℄. The Ramanujan-Peterson-Selberg onje tureasserts that

0

is admissible.

Denition 3.3Let

T : R

3

_{→ R}

+

and

(M, N, C) ∈ (R \ {0})

3

, we say that a smooth fun tion

h : R

3

_{→ R}

3

satises the property

P(T ; M, N, C)

if there exists a real number

K > 0

su h that

∀(i, j, k) ∈ N

3

, ∀(x

1

, x

2

, x

3

) ∈



M

2

, 2M



×



N

2

, 2N



×



C

2

, 2C



,

x

i

₁

x

j

₂

x

k

₃

∂

i+j+k

_h

∂x

i

₁

∂x

j

₂

∂x

₃

k

(x

1

, x

2

, x

3

) 6 KT (M, N, C)

1 +

√

M N

C

!

i+j+k

.

With this denition in mind, we areable to write the following proposi-tion whi h isspe ial aseofalarge sieve inequalityadaptedfromtheoneof Deshouillers&Iwanie [5,Theorem9℄byBlomer,Har os &Mi hel[2, The-orem 4℄.

Proposition 3.4 Let

q

be some positive integer. Let

M, N, C > 1

and

g

be a smoothfun tionsatisfyingproperty

P(1; M, N, C)

. Considertwo sequen es of omplex numbers

(a

m

)

m∈[M/2,2M]

and

(b

n

)

n∈[N/2,2N]

. If

θ

is admissible and

M N ≪ C

2

then

X

c>1

q|c

X

m>1

X

n>1

a

m

b

n

S(m, ±n; c)

c

g(m, n; c)

≪

ε

(qM N C)

ε



C

2

M N



θ



1 +

M

q



1/2



1 +

N

q



1/2

kak

2

kbk

2

(3.1) for any

ε > 0

.

We shall use a test fun tion. For any

ν > 0

let us dene

S

ν

(R)

as the spa e ofevenS hwartz fun tion

Φ

whose Fourier transform

b

Φ(ξ) := F[x 7→ Φ(x)](ξ) :=

Z

R

Φ(x)e(−xξ) dx

is ompa tlysupportedin

[−ν, +ν]

. ThankstotheFourierinversionformula:

Φ(x) =

Z

R

b

Φ(ξ)e(xξ) dx = F[ξ 7→ b

Φ(ξ)](−x),

(3.2)

su hafun tion

Φ

an be extendedtoanentire evenfun tion whi hsatises

∀s ∈ C,

Φ(s) ≪

n

exp (ν|ℑm s|)

(1 + |s|)

n

(3.3)

for anyinteger

n > 0

.The versionof the large sieve inequalitywe shall use several timesinthis paperisthenthe following.

Corollary 3.5 Let

q

be some prime number,

k

1

, k

2

> 0

be some integers,

α

1

, α

2

, ν

be some positive real numbers and

Φ ∈ S

ν

(R)

. Let

h

be some smooth fun tion satisfyingproperty

P(T ; M, N, C)

for any

1 6 M 6 q

k

1

α

1

ν

,

1 6 N 6 q

k

2

α

2

ν

and

C > q

. Let

(a

p

)

p∈P

p6q

α1ν

and

(b

p

)

p∈P

p6q

α2ν

be some omplex numbers sequen es. If

θ

isadmissible and

ν 6 2 /(k

1

α

1

+ k

2

α

2

)

then

X

c>1

q|c

X

p

1

∈P

p

1

∤q

X

p

2

∈P

p

2

∤q

a

p

1

b

p

2

S(p

k

1

1

, p

k

2

2

; c)

c

h



p

k

1

1

, p

k

2

2

; c



b

Φ



log p

1

log(q

α

1

₎



b

Φ



log p

2

log(q

α

2

₎



≪ q

ε

X

♯

16M 6q

να1k1

16N 6q

να2k2

C>q/2

1 +

s

M

q

!

1 +

s

N

q

! 

C

2

M N



θ

T (M, N, C)kak

2

kbk

2

(3.4) where

♯

indi atesthat the sum ison powers of

√

2

. The onstant implied by the symbol

≪

depends atmost on

ε

,

k

1

,

k

2

,

α

1

,

α

2

and

ν

.

Proof. Dene

(ba

m

)

m∈N

,



bb

n



n∈N

and

g(m, n; c)

by

ba

m

:= a

_m

1/k1

1

_P

k1

(m) 1

[1,q

να1k1

]

(m)

(3.5)

bb

n

:= b

_n

1/k1

1

_P

k1

(n) 1

[1,q

να1k1

]

(n)

(3.6)

g(m, n; c) := h(m, n, c)b

Φ



log m

log(q

α

1

k

1

)



b

Φ



log n

log(q

α

2

k

2

)



.

(3.7)

Using a smooth partition of unity,asdetailed inŸA.1, we need to evaluate

X

♯

16M 6q

να1k1

16N 6q

να2k2

C>q/2

T (M, N, C)

X

c>1

q|c

X

m>1

X

n>1

ba

m

bb

n

S(m, n; c)

c

g

M,N,C

(m, n; c)

T (M, N, C)

.

(3.8)

Sin e

ν 6 2 /(α

1

k

1

+ α

2

k

2

)

,therst summation isrestri ted to

M N ≪ C

2

hen e, using proposition3.4, thequantityin(3.8) is

≪ kak

2

kbk

2

q

ε

X

♯

16M 6q

να1k1

16N 6q

να2k2

C>q/2

T (M, N, C)

1 +

s

M

q

!

1 +

s

N

q

! 

C

2

M N



θ

.

(3.9)



3.2. Riemann's expli it formula for symmetri power

L

-fun tions. Inthisse tion,wegiveananalogofRiemann-vonMangoldt'sexpli itformula forsymmetri power

L

-fun tions. Beforethat,letusre allsomepreliminary fa tsonzerosofsymmetri power

L

-fun tionswhi h anbefoundinse tion 5.3 of [17℄. Let

r > 1

and

f ∈ H

∗

κ

(q)

for whi h hypothesis

Nice(r, f )

holds. All the zeros of

Λ(Sym

r

_{f, s)}

are inthe riti al strip

{s ∈ C: 0 < ℜe s < 1}

. Themultiset ofthezerosof

Λ(Sym

r

_{f, s)}

ountedwithmultipli ities isgiven by

n

ρ

(j)

_f,r

= β

_f,r

(j)

+ iγ

(j)

_f,r

_{: j ∈ E(f, r)}

o

where

E(f, r) :=

(

Z

if

Sym

r

_f

isodd

Z \ {0}

if

Sym

r

_f

iseven. and

β

_f,r

(j)

_{= ℜe ρ}

(j)

_f,r

,

γ

_f,r

(j)

_{= ℑm ρ}

(j)

_f,r

for any

j ∈ E(f, r)

. Weenumeratethezeros su h that (1) the sequen e

j 7→ γ

(j)

f,r

isin reasing (2) we have

j > 0

ifand only if

γ

(j)

f,r

>

0

(3) we have

ρ

(−j)

f,r

= 1 − ρ

(j)

f,r

. Note thatif

ρ

(j)

f,r

isa zeroof

Λ(Sym

r

_{f, s)}

then

ρ

(j)

f,r

,

1 − ρ

(j)

f,r

and

1 − ρ

(j)

f,r

are

alsosomezerosof

Λ(Sym

r

_{f, s)}

. Inaddition,rememberthatif

Sym

r

_f

isodd then the fun tional equation of

L(Sym

r

_{f, s)}

s = 1/2

providesatrivialzerodenotedby

ρ

(0)

f,r

. It anbeshown[17,Theorem 5.8℄ that thenumber ofzeros

Λ(Sym

r

_{f, s)}

satisfying

|γ

(j)

f,r

| 6 T

is

T

π

log



q

r

T

r+1

(2πe)

r+1



+ O (log(qT ))

(3.10)

as

T > 1

goestoinnity. WestatenowtheGeneralisedRiemannHypothesis whi h is themain onje ture about the horizontal distribution of the zeros of

Λ(Sym

r

_{f, s)}

inthe riti alstrip.

Hypothesis

GRH(r)

 For any prime number

q

and any

f

in

H

∗

κ

(q)

, all

the zeros of

Λ(Sym

r

_{f, s)}

lieon the riti al line

{s ∈ C: ℜe s = 1/2}

namely

β

_r,f

(j)

= 1/2

for any

j ∈ E(f, r)

.

Remark 3.6We donot usethis hypothesisinour proofs.

Under hypothesis

GRH(r)

, it an be shown that the number of zeros of the fun tion

Λ(Sym

r

_{f, s)}

satisfying

|γ

(j)

f,r

| 6 1

is given by

1

π

log (q

r

_{)(1 + o(1))}

as

q

goes to innity. Thus, thespa ing between two onse utive zeros with imaginary partin

[0, 1]

is roughlyofsize

2π

log (q

r

₎

.

(3.11)

We aim at studying thelo al distribution of the zeros of

Λ(Sym

r

_{f, s)}

ina neighborhood ofthe realaxis ofsize

1/ log q

r

sin ein su ha neighborhood, we expe t to at h only few zeros (but without being able to say that we at honly one

2

). Hen e, we normalisethe zeros bydening

b

ρ

(j)

_f,r

:=

log (q

r

₎

2iπ



β

_f,r

(j)

₋

1

2

+ iγ

(j)

f,r



.

Note that

b

ρ

(−j)

_f,r

_{= −b}

ρ

(j)

_f,r

.

Denition 3.7 Let

f ∈ H

∗

κ

(q)

for whi h hypothesis

Nice(r, f )

holds and let

Φ ∈ S

ν

(R)

. The one-leveldensity (relatively to

Φ

)of

Sym

r

_f

is

D

1,q

[Φ; r](f ) :=

X

j∈E(f,r)

Φ



ρ

_b

(j)

_f,r



.

(3.12)

To study

D

1,q

[Φ; r](f )

for any

Φ ∈ S

ν

(R)

, we transform this sum over zeros into a sum over primes in the next proposition. In other words, we establish an expli it formula for symmetri power

L

-fun tions. Sin e the proofis lassi al,wereferto[18,Ÿ4℄or[9,Ÿ2.2℄whi hpresentamethodthat hasjust tobe adaptedto our setting.

2

Proposition 3.8 Let

r > 1

and

f ∈ H

∗

κ

(q)

for whi h hypothesis

Nice(r, f )

holds and let

Φ ∈ S

ν

(R)

. We have

D

1,q

[Φ; r](f ) = E[Φ; r]+P

q

1

[Φ; r](f )+

r−1

X

m=0

(−1)

m

P

_q

2

[Φ; r, m](f )+O



1

log (q

r

₎



where

E[Φ; r] := b

Φ(0) +

(−1)

r+1

2

Φ(0),

P

_q

1

_{[Φ; r](f ) := −}

2

log (q

r

₎

X

p∈P

p∤q

λ

f

(p

r

)

log p

√

_p

Φ

b



log p

log (q

r

₎



,

P

_q

2

_{[Φ; r, m](f ) := −}

2

log (q

r

₎

X

p∈P

p∤q

λ

f



p

2(r−m)

 log p

p

Φ

b



2 log p

log (q

r

₎



for any integer

m ∈ {0, . . . , r − 1}

.

3.3. Contribution of the old forms. In this short se tion, we prove the following useful lemmas.

Lemma 3.9 Let

p

1

and

p

2

6= q

be some prime numbers and

a

1

,

a

2

,

a

be some nonnegative integers. Then

X

ℓ|q

∞

∆

₁

(ℓ

2

p

a

1

1

, p

a

2

2

q

a

)

ℓ

≪

1

q

a/2

the implied onstant depending onlyon

a

1

and

a

2

.

Proof. Usingproposition 2.1and thefa tthat

H

κ

(1) = H

∗

κ

(1)

,wewrite

∆

₁

(ℓ

2

p

a

1

1

, p

a

2

2

q

a

) =

X

h

f ∈H

∗

κ

(1)

λ

_f

(ℓ

2

p

a

1

1

)λ

f

(p

a

2

2

q

a

)

(3.13)

≪

X

h

f ∈H

∗

κ

(1)

|λ

f

(ℓ

2

p

a

1

1

)| · |λ

f

(p

a

2

2

)| · |λ

f

(q

a

)|.

(3.14)

By Deligne's bound (2.3)wehave

|λ

f

(ℓ

2

p

a

1

1

)| · |λ

f

(p

a

2

2

)| 6 τ(ℓ

2

p

a

1

1

)τ (p

a

2

2

) 6 (a

1

+ 1)(a

2

+ 2)τ (ℓ

2

).

(3.15)

By the multipli ativity relation (2.4) and the valueof thesign of the fun -tional equation(2.15) , we have

|λ

f

(q

a

)| ≪

1

q

a/2

.

(3.16) We obtain theresult byreporting (3.16) and (3.15) in (3.14) and by using (2.16) and

X

ℓ|q

∞

τ (ℓ

2

₎

ℓ

=

1 + 1/q

(1 − 1/q)

2

≪ 1.