Lower order terms for the one-level densities of symmetric power $L$-functions in the level aspect

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symmetric power L-functions in the level aspect

Guillaume Ricotta, Emmanuel Royer

To cite this version:

Guillaume Ricotta, Emmanuel Royer. Lower order terms for the one-level densities of symmetric

power L-functions in the level aspect. Acta Arithmetica, Instytut Matematyczny PAN, 2010, 141 (2),

pp.153-170. �hal-00288656�

GUILLAUMERICOTTAANDEMMANUELROYER

Abstra t. In[12 ℄,theauthorsdetermined,amongotherthings,themaintermsfor

theone-leveldensities forlow-lyingzerosofsymmetri power

L

-fun tionsinthelevel aspe t. In this paper, the lower orderterms of these one-level densities are found. The ombinatorial di ulties,whi hshouldariseinsu h ontext,are drasti ally

re-du edthanks to Chebyshevpolynomials,whi hare the hara tersof theirredu ible representations of

SU(2)

.

Contents

1. Introdu tion and statementof theresult 1

1.1. Des ription of the familiesof

L

-fun tionsstudied 1

1.2. One-level densitiesof thesefamilies 3

2. Chebyshev polynomials and He ke eigenvalues 4

3. Riemann's expli itformulafor symmetri power

L

-fun tions 6

4. Proofof Theorem A 8

Appendix A. Some ommentson an aestheti identity 10

Appendix B. S.J. Miller'sidentityand Cheby hev polynomials 11

Referen es 11

A knowledgements The rst author is nan ed by the ANR proje t Aspe ts

Arithmé-tiquesdes Matri es Aléatoires et duChaos Quantique. He would like tothank the

Uni-versity of Nottingham,where this work has been nished, for its hospitality. The se ond

one is supported by the ANR proje t Modunombres. Both authors would like to thank

the anonymousreferee of [12℄ for suggesting them this problem.

1. Introdu tion and statement of the result

1.1. Des ription ofthefamiliesof

L

-fun tionsstudied. Thepurposeofthispaper isto ompute the lowerorder termsofsome parti ular statisti sasso iated to low-lying

zerosofseveralfamiliesofsymmetri power

L

-fun tionsinthelevelaspe t: theone-level densities. First of all, we give a short des ription of these families. To any primitive

holomorphi uspform

f

of primelevel

q

and even weight 1

κ > 2

(see [12 ,Ÿ 2.1℄ forthe automorphi ba kground) say

f ∈ H

∗

κ

(q)

, one an asso iate its

r

-th symmetri power

L

-fun tion denoted by

L(Sym

r

_{f, s)}

for any integer

r > 1

. It is given by the following absolutely onvergent andnon-vanishingEuler produ t ofdegree

r + 1

on

ℜe s > 1

L(Sym

r

f, s) =

Y

p∈P

L

p

(Sym

r

f, s)

Date:VersionofJune18,2008. 1

Inthispaper,theweight

κ

isaxed evenintegerand thelevel

q

goestoinnityamongtheprime numbers.

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

. From now on,

α

f

(p)

,

β

f

(p)

are the Satake parameters of

f

at the prime number

p

and

(λ

f

(n))

n>1

is its sequen e of He ke eigenvalues, whi h is arithmeti ally normalised:

λ

f

(1) = 1

and

|λ

f

(p)| 6 2

for any prime

p

. We also dene [1, (3.16) and (3.17)℄ a lo alfa tor at

∞

whi h is given by a produ t of

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)

(1.1) if

r

isodd and

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))

(1.2) if

r

iseven where

µ

κ,r

=

(

1

if

r(κ − 1)/2

isodd,

0

otherwise. The ompleted

L

-fun tionis dened by

Λ(Sym

r

f, s) = (q

r

)

s/2

L

∞

(Sym

r

f, s)L(Sym

r

f, s)

and

q

r

isthearithmeti ondu tor. Wewillneedsome ontrolontheanalyti behaviour

ofthisfun tion. Unfortunately,su hinformationisnot urrentlyknowninallgenerality.

We sumup our main assumption inthefollowing statement.

Hypothesis

Nice(r, f )

 The fun tion

Λ (Sym

r

_{f, s)}

is a ompleted

L

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

•

it an be extended to an 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)}

where the sign

ε (Sym

r

_{f ) = ±1}

of the fun tionalequation isgiven by

ε (Sym

r

f ) =

(

+1

if

r

iseven

,

ε

f

(q) × ε(κ, r)

otherwise (1.3) 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) = −

√

_qλ

f

(q) = ±1

.

Remark 1Hypothesis

Nice(r, f )

is known for

r = 1

(E. He ke [3, 4 , 5℄),

r = 2

thanks to theworkofS.Gelbartand H.Ja quet[2 ℄and

r = 3, 4

fromtheworksofH.Kimand F. Shahidi[9 , 8,7 ℄.

We aim at studying the lower order terms of the one-level density for the family of

L

-fun tions given by

[

q

prime

{L(Sym

r

f, s), f ∈ H

κ

∗

(q)}

1.2. One-leveldensities of these families. Thepurposeofthisworkistodetermine

thelowerordertermsoftheone-leveldensitiesasso iatedtothesefamiliesof

L

-fun tions. Let us give the statement of our result, in whi h

ν

is a positive real number,

Φ

is an even S hwartz fun tion,whose Fourier transform

Φ

b

is ompa tly supported in

[−ν, +ν]

(denoted by

Φ ∈ S

ν

(R)

) and

f

is a primitive holomorphi usp form of prime level

q

and even weight

κ > 2

for whi h hypothesis

Nice(r, f )

holds

2

. We refer to [12 , Ÿ 2.2℄

fortheprobabilisti ba kground. Notethat, thankstoFourier inversionformula,su h a

fun tion

Φ

an be extendedto an entire even fun tion whi h satises

∀s ∈ C,

Φ(s) ≪

n

exp (ν|ℑm s|)

(1 + |s|)

n

(1.4)

for any integer

n > 0

. Theone-level density (relativelyto

Φ

) of

Sym

r

_f

isdened 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



wherethesumisoverthenon-trivialzeros

ρ

of

L(Sym

r

_{f, s)}

repeatedwithmultipli ities.

Theasymptoti expe tation oftheone-leveldensityis bydenition

lim

q

prime

q→+∞

X

f ∈H

∗

κ

(q)

ω

q

(f )D

1,q

[Φ; r](f )

where

ω

q

(f ) =

Γ(κ−1)

(4π)

κ−1

_{hf,f i}

_q

isthe harmoni weight of

f

. Beforestatingour result,let us dene thefollowing onstants:

C

PNT

=



1 +

Z

+∞

1

θ(t) − t

t

2

dt



,

(1.5)

C =

X

p∈P

log p

p

3/2

_{− p}

,

(1.6)

C

∞

= −(r + 1) log π + C

Γ

(1.7)

where

θ

is the rstChebyshevfun tion:

θ(t) =

X

p

prime

p6t

log p,

C

_Γ

=

X

06a6(r−1)/2



Γ

′

Γ

 

1

4

+

(2a + 1)(κ − 1)

4



+



Γ

′

Γ

 

1

4

+

1

2

+

(2a + 1)(κ − 1)

4



(1.8) if

r

isodd and

C

Γ

=



Γ

′

Γ

 

1

4

+

µ

κ,r

2



+

X

16a6r/2



Γ

′

Γ

 

1

4

+

a(κ − 1)

2



+



Γ

′

Γ

 

1

4

+

1

2

+

a(κ − 1)

2



(1.9) if

r

iseven.

TheoremA Let

r > 1

be anyintegerand

ε = ±1

. We assumethathypothesis

Nice(r, f )

holds for any prime number

q

and any primitive holomorphi usp form of level

q

and even weight

κ > 2

. Let

ν

1,max

(r, κ, θ

0

) =



1 −

1

2(κ − 2θ

0

)



2

r

2

2

-with

θ

0

= 7/64

. If

ν < ν

1,max

(r, κ, θ

0

)

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



b

Φ(0) +

(−1)

r+1

2

Φ(0)



+



C

∞

− 2(−1)

r

C

PNT

− 2δ

2|r

C

 b

Φ(0)

log q

r

+ O



1

log

3

(q

r

₎



.

Remark 2The main terms of the asymptoti expe tation of these one-level densities

have already been found in [12 ℄ (see Theorem B). The new information is the lower

orderterms namelyterms ofsize

1/ log (q

r

₎

.

Remark 3Note that

θ

0

= 7/64

is the best known approximation towards Ramanujan-Peterson-Selberg's onje ture (see [12 , hypothesis

H

2

(θ)

page 16℄) thanks to theworks of H.Kim,F. Shahidi andP.Sarnak ([8 ,7℄). The value

θ = 0

is expe ted.

Remark 4 It is lear from the proof of Theorem A that the same result holds for the

signed familieswiththesame restri tion onthe support asin[12℄.

Remark 5 Theparti ular ase

r = 1

hasalreadybeen investigated byS.J. Miller[11 ℄. NotationWe write

P

fortheset ofprimenumbersandthe mainparameterinthispaper isa primenumber

q

, whose name is the level, whi h goes to innityamong

P

. Thus, if

f

and

g

aresome

C

-valuedfun tionsofthereal variablethenthenotations

f (q) ≪

A

g(q)

or

f (q) = O

A

(g(q))

mean that

|f(q)|

issmallerthan a  onstant whi honlydepends on

A

times

g(q)

at least for

q

a large enough prime number.

2. Chebyshev polynomials and He ke eigenvalues

Re allthatthegeneral fa tsaboutholomorphi uspforms anbefoundin[12 ,Ÿ2.1℄.

Let

p 6= q

aprime number and

f ∈ H

∗

κ

(q)

. Denote by

χ

St

the hara ter ofthe standard representation

St

of

SU(2)

. Bythework of Deligne, thereexists

θ

f,p

∈ [0, π]

su h that

λ

_f

(p) = χ

St



e

iθ

f,p

₀

0

e

−iθ

f,p



.

Moreoverthe multipli ativityrelation reads

λ

f

(p

ν

) = χ

Sym

ν



e

iθ

f,p

₀

0

e

−iθ

f,p



= X

ν



χ

St



e

iθ

f,p

₀

0

e

−iθ

f,p



= X

ν

(λ

f

(p))

(2.1)

where

χ

Sym

ν

isthe hara terofthe irredu iblerepresentation

Sym

ν

_St

of

SU(2)

andthe polynomials

X

ν

aredened bytheir generatingseries

X

ν>0

X

ν

(x)t

ν

=

1

1 − xt + t

2

.

(2.2)

Theyareequivalentely dened by

X

ν

(2 cos θ) =

sin ((ν + 1)θ)

sin (θ)

.

(2.3)

These polynomials are known as Chebyshev polynomials of se ond kind. Ea h

X

ν

has degree

ν

, is even if

ν

is even and odd otherwise. The family

(X

ν

)

ν>0

is a basis for the polynomialve tor spa e

Q[T ]

,orthonormal withrespe tto theinnerprodu t

hP, Qi

ST

=

1

π

Z

2

−2

P (x)Q(x)

r

1 −

x

2

4

dx.

Thefollowing propositionlists Chebyshevpolynomials'neededproperties forthis work.

•

If

̟ > 0

is any integerthen

X

_r

̟

=

r̟

X

j=0

x(̟, r, j)X

j

(2.4) 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.

(2.5) In parti ular,

x(̟, r, j) =







0

if

j ≡ r̟ + 1 (mod 2),

(

̟

̟/2

)

1+̟/2

if

̟

is even,

r = 1

and

j = 0

. (2.6)

•

If

α

isa omplex number of norm

1

and

n > 0

isan integer then

α

n

+ α

−n

=











2X

0

(α + α

−1

)

if

n = 0

,

X

1

(α + α

−1

)

if

n = 1

,

X

n

(α + α

−1

) − X

n−2

(α + α

−1

)

otherwise. (2.7)

•

If

α

isa omplex number of norm

1

and

r, n > 1

are someintegers then

S(α; n, r) =

r

X

j=0

α

n(2j−r)

= δ

_2|r

+

X

16j6r

j≡r

(mod 2)



α

jn

+ α

−jn



(2.8)

=

X

06j6r

j≡r

(mod 2)



X

jn

(α + α

−1

) − X

jn−2

(α + α

−1

)



(2.9)

= X

r

(α

n

+ α

−n

)

(2.10) where

X

−1

= X

−2

= 0

by onvention.

•

If

r > 1

and

n > 1

are someintegers then

X

06j6r

j≡r

(mod 2)

[X

jn

− X

jn−2

] =

r

X

j=0

(−1)

j

X

_n−2

j

X

_n(r−j)

(2.11) where

X

−1

= X

−2

= 0

by onvention.

•

If

ℓ > 0

isan integerthen

X

ℓ

=

X

06u6ℓ

u≡ℓ

(mod 2)

(−1)

(ℓ−u)/2



(ℓ + u)/2

u



T

u

.

(2.12)

Proof of proposition2.1. Therst point follows fromthefa tthat

X

̟

r

isanpolynomial ofdegree

r̟

,whi hisevenif

r̟

isevenandoddotherwise. Thus,(2.4)istheexpansion ofthis polynomialinthe orthonormal basis

(X

j

)

06j6r̟

. The se ondpoint follows from theequality

2 cos (nθ) sin (θ) = sin ((n + 1)θ) − sin ((n − 1)θ).

If

α = exp (iθ)

thenthis equality ombined with(2.3) leadto

2 cos (nθ) = X

n

(2 cos θ) − X

n−2

(2 cos θ),

whi histhe desiredresultsin e

2 cos θ = α + α

−1

and

2 cos (nθ) = α

n

_{+ α}

−n

. Thethird

pointis adire t onsequen e of these ondone, ofthedire t omputation

S(α; n, r) =

α

n(r+1)

_{− α}

−n(r+1)

and of

X

r

(α

n

+ α

−n

) = X

r

(2 cos (nθ)) =

α

n(r+1)

_{− α}

−n(r+1)

α

n

_{− α}

−n

if

α = exp (iθ)

. The fourthpoint is easily dedu edfromthefa tthat

S(α; n, r) =

r

X

j=0

(−1)

j

X

_n−2

j

(α + α

−1

)X

n(r−j)

(α + α

−1

)

for any omplex number

α

of norm

1

. Letus prove theprevious equality. A ording to [13, Page 727, rstand se ondequations℄,

X

r>0

X

nr

(α + α

−1

)t

r

=



1 + X

n−2

(α + α

−1

)t

 X

r>0

X

r

(α

n

+ α

−n

)t

r

.

Asa onsequen e,

X

nr

(α + α

−1

) = X

r

(α

n

+ α

−n

) + X

n−2

(α + α

−1

)X

r−1

(α

n

+ α

−n

),

whi himplies

X

r

(α

n

+ α

−n

) =

r

X

j=0

(−1)

j

X

_n−2

j

(α + α

−1

)X

n(r−j)

(α + α

−1

).

Thelast point isobtained by developping (2.2) asan entire series in

x

.



3. Riemann'sexpli it formula for symmetri power

L

-fun tions Tostudy

D

1,q

[Φ; r](f )

forany

Φ ∈ S

ν

(R)

,wetransformthissumoverzerosintoasum overprimes inthenextproposition. Inotherwords,weestablishanexpli itformulafor

symmetri power

L

-fun tions.

Proposition 3.1 Let

r > 1

and

f ∈ H

∗

κ

(q)

for whi hhypothesis

Nice(r, f )

holds andlet

Φ ∈ S

ν

(R)

. We have

D

1,q

[Φ; r](f ) =



b

Φ(0) +

(−1)

r+1

2

Φ(0)



+

Φ(0)

b

log (q

r

₎



C

∞

+ 2(−1)

r+1

C

PNT

− 2δ

2|r

C



+ P

_q

1

[Φ; r](f ) +

r−1

X

m=0

(−1)

m

P

_q

2

[Φ; r, m](f ) + P

_q

3

[Φ; r](f ) + O



1

log

3

(q

r

₎



where

C

PNT is dened in (1.5) ,

C

in (1.6) ,

C

∞

in (1.7) whereas

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

₎



P

_q

3

_{[Φ; r](f ) = −}

2

log (q

r

₎

X

p∈P

p∤q

X

n>3









X

16j6r

j≡r

(mod 2)

λ

f

(p

jn

) − λ

f

(p

jn−2

)









log p

p

n/2

Φ

b



n log p

log (q

r

₎



for any integer

m ∈ {0, . . . , r − 1}

. Proof of proposition3.1. Let

G(s) = Φ



log (q

r

)

2iπ



s −

1

₂



.

From [6,eq. (4.11) and (4.14)℄we get

D

1,q

[Φ; r](f ) = b

Φ(0) − (r + 1)

log π

log q

r

Φ(0)

b

−

_{log q}

2

_r

X

p∈P

+∞

X

m=1





r

X

j=0

α

f

(p)

jm

β

f

(p)

(r−j)m



 b

Φ



m log p

log q

r



log p

p

m/2

+

Φ(0)

b

log q

r

r

X

j=0

Γ

′

Γ



1

4

+

µ

j

2



+ O



1

log

3

q



.

(3.1)

Letusfo usonthethirdterm in(3.1) . Notthatthe ontributionoftheprime

q

isgiven by

−

2

_r

+∞

X

m=1



_λ

f

(q)

r

√

_q



m

b

Φ

m

r



≪

1

q

(r+1)/2

and for

p 6= q

we use

r

X

j=0

α

f

(p)

jm

β

f

(p)

(r−j)m

= S (α

f

(p); m, r)

withthenotation of (2.8) . We obtain

S (α

f

(p); 1, r) = X

r

α

f

(p) + α

f

(p)

−1

= λ

f

(p

r

)

a ordingto (2.1) and

S (α

f

(p); 2, r) =

X

06j6r

j≡r

(mod 2)

X

2j

α

f

(p) + α

f

(p)

−1

− X

2j−2

α

f

(p) + α

f

(p)

−1

=

r

X

j=0

(−1)

j

X

_2(r−j)

α

f

(p) + α

f

(p)

−1

(

f.(2.11)

)

=

r−1

X

m=0

(−1)

m

λ

f



p

2(r−m)



_{+ (−1)}

r

.

Asa onsequen e,

X

p∈P

p6=q

+∞

X

m=1





r

X

j=0

α

f

(p)

jm

β

f

(p)

(r−j)m



 =

X

p∈P

p∤q

λ

f

(p

r

) log p

p

1/2

Φ

b



log p

log (q

r

₎



+

X

p∈P

p∤q

r−1

X

m=0

(−1)

m

λ

f



p

2(r−m)



!

log p

p

Φ

b



log (p

2

₎

log (q

r

₎



+ (−1)

r

X

p∈P

p∤q

log p

p

Φ

b



log (p

2

)

log (q

r

₎



+

X

p∈P

p∤q

X

n>3

S (α

f

(p); n, r)

log p

p

n/2

Φ

b



log (p

n

)

log (q

r

₎



.

(3.2)

summation, thisterm equals, up to

O(q

−0.9

₎

,

(−1)

r

Z

+∞

1

θ(t)

t

2



b

Φ



2 log t

log (q

r

₎



−

_{log (q}

2

_r

₎

Φ

b

′



2 log t

log (q

r

₎



dt := S

3

.

Then,

S

3

= (−1)

r

Z

+∞

1



b

Φ



2 log t

log (q

r

₎



−

_{log (q}

2

_r

₎

Φ

b

′



2 log t

log (q

r

₎



dt

t

+ (−1)

r

Z

+∞

1

θ(t) − t

t



b

Φ



2 log t

log (q

r

₎



−

_{log (q}

2

_r

₎

Φ

b

′



2 log t

log (q

r

₎



dt

t

.

Sin e

Φ(u) = b

b

Φ(0) + O(u

2

₎

and

Φ

b

′

_{(u) ≪ |u|}

,we get

S

3

= (−1)

r

log (q

r

₎

2

Z

+∞

0

b

Φ(u) du − (−1)

r

Z

+∞

0

b

Φ

′

_{(u) du + (−1)}

r

Φ(0)

b

Z

+∞

1

θ(t) − t

t

2

dt

+ O



1

log

2

(q

r

₎



and nally

S

3

= (−1)

r

log (q

r

)

4

Φ(0) + (−1)

r

_Φ(0)

_b



_{1 +}

Z

+∞

1

θ(t) − t

t

2

dt



+ O



1

log

2

(q

r

₎



.

We nallytake areofthe fourth term of (3.2). A ording to (2.1) and (2.8) ,we have

S (α

f

(p); n, r) = δ

2|r

+

X

16j6r

j≡r

(mod 2)



λ

f

(p

jn

) − λ

f

(p

jn−2

)



.

Onemayremark that

X

p∈P

p∤q

X

n>3

log p

p

n/2

Φ

b



n log p

log (q

r

₎



=

X

p∈P

X

n>3

log p

p

n/2

Φ (0) + O

b



1

log

3

(q

r

₎



sin e

Φ(u) = b

b

Φ(0) + O(u

2

₎

. Then, we easily get

X

p∈P

X

n>3

log p

p

n/2

=

X

p∈P

log p

p

3/2

_{− p}

.



4. Proof of Theorem A

Theaim of thispartis to determine anasymptoti expansion of

X

f ∈H

∗

κ

(q)

ω

q

(f )D

1,q

[Φ; r](f ) = E

h

q

(D

1,q

[Φ; r]) .

A ording to proposition3.1and theproofof [12 , eq. (4.6) and (4.7)℄, if

ν <



1 −

1

2(κ − 2θ)



2

r

2

(4.1) then

E

h

_q

(D

1,q

[Φ; r]) =



b

Φ(0) +

(−1)

r+1

2

Φ(0)



+

Φ(0)

b

log (q

r

₎



C

∞

+ 2(−1)

r+1

C

PNT

− 2δ

2|r

C



+ E

h

_q

P

_q

3

[Φ; r](f )

+ O



1

log

3

(q

r

₎



.

(4.2)

The rst term in (4.2) is the main term given in Theorem A. We now estimate the

penultemate term of (4.2)via thetra e formulagiven in[12 ,Proposition 2.2℄:

E

h

_q

P

_q

3

[Φ; r]

= P

3

_q,new

[Φ; r] + P

3

_q,old

[Φ; r]

(4.3) where

P

3

_q,new

_{[Φ; r] = −}

2

log (q

r

₎

X

p∈P

p∤q

X

n>3









X

16j6r

j≡r

(mod 2)

∆

q

(p

jn

, 1) − ∆

q

(p

jn−2

, 1)









log p

p

n/2

Φ

b



n log p

log (q

r

₎



and

P

3

_q,old

[Φ; r] =

2

q log (q

r

₎

X

ℓ|q

∞

1

ℓ

X

p∈P

p∤q

X

n>3









X

16j6r

j≡r

(mod 2)

∆

1

(p

jn

ℓ

2

, 1) − ∆

1

(p

jn−2

ℓ

2

, 1)









log p

p

n/2

Φ

b



n log p

log (q

r

₎



.

For

m 6= 1

we have

∆

k

(m, 1) = 2πi

κ

X

c>1

k|c

S(m, 1; c)

c

J

κ−1



4π

√

m

c



where

S(m, 1; c)

is a Kloosterman sum. Let us estimate the new part whi h an be written as

P

3

_q,new

_{[Φ; r] = −}

2(2πi

κ

₎

log (q

r

₎

X

16j6r

j≡r

(mod 2)

X

n>3

P

3

_q,new

_{[Φ; r, jn] − P}

3

_q,new

_{[Φ; r, jn − 2]}

where

P

3

_q,new

[Φ; r, k] =

X

p∈P

p6=q

log p

p

n/2

Φ

b

log p

log q

r/n

!

X

c>1

q|c

S(p

k

, 1; c)

c

J

κ−1

4π

p

p

k

c

!

.

(4.4)

By [12,lemma 3.10℄,the

c

-sum in(4.4) isbounded by

τ (q)

√

_q















 √

p

k

q



1/2

if

p > q

2/k

,

 √

p

k

q



κ−1

otherwise. We dedu e

X

n>3

P

3

_q,new

_{[Φ; r, jn] ≪}

τ (q)

q

κ−1/2

X

n>3

X

p6q

rν/n

1

p

n/2

p

rn(κ−1)/2

_{log p}

≪

τ (q)

q

κ−1/2

X

36n6νr log q/log 2

1

n

q

νr[((κ−1)r−1)n/2+1]/n

≪

τ (q)

q

κ−1/2

q

νr[(κ−1)r−1]/2

_q

νr/3

_{log log(3q)}

≪

1

q

1/2

assoon as

ν < 2/r

2

(and in parti ular if (4.1) is satised). We make thesame

ompu-tationsfor

jn − 2

andnd thenthat

P

3

q,new

[Φ; r, k]

is anadmissible error term. Theold partis

P

3

_q,old

[Φ; r] =

2(2πi

κ

₎

q log (q

r

₎

X

16j6r

j≡r

(mod 2)

X

n>3

P

3

_q,old

_{[Φ; r, jn] − P}

3

_q,old

_{[Φ; r, jn − 2]}

where

P

3

_q,old

[Φ; r, k] =

X

p∈P

p6=q

log p

p

n/2

Φ

b



log p

log q

r/n

 X

ℓ|q

∞

1

ℓ

∆

1

(p

k

_ℓ

2

_{, 1).}

From [12,eq (3.2) and(3.3)℄ we have

X

ℓ|q

∞

1

ℓ

∆

1

(p

k

_ℓ

2

_{, 1) 6 2(k + 1)}

sothat

X

n>3

P

3

_q,old

_{[Φ; r, jn] ≪ 1}

and similaryfor

P

3

q,old

[Φ; r, jn − 2]

. Finally

E

h

q

P

q

3

[Φ; r]

enters the

O(1/ log

3

_q

r

₎

term.

Appendix A. Some omments onan aestheti identity

Itis possible to prove onindu tion on

k

0

>

1

thefollowing equalityin

Q

[T ]

:

X

2k

0

− X

2k

0

−2

=

k

_X

0

−1

j=0

X

16k

j

<k

j−1

<···<k

1

<k

0

(−1)

j

"

_j−1

Y

i=0



2k

i

k

i

− k

i+1

# 

T

2k

j

₋



2k

j

k

j



.

(A.1) Asa onsequen e,if

K > 1

then

X

2K+1

− X

2K−1

= (−1)

K

T



1 +

X

16k

0

6K

(−1)

k

0

_X

2k

0

− X

2k

0

−2



 .

(A.2)

Now, use(2.4) with

r = 1

(sothat

X

1

= T

)to getfrom (A.1) theequality

X

2k

0

− X

2k

0

−2

=

k

_X

0

−1

j=0

X

16k

j

<k

j−1

<···<k

1

<k

0

(−1)

j

"

_j−1

Y

i=0



_2k

i

k

i

− k

i+1

#





2k

j

X

ℓ=0

x(2k

j

, 1, ℓ)X

ℓ

−



_2k

j

k

j



X

0





and ompare the oe ientsof

X

0

to obtain, thanksto (2.6)the equality

k

0

−1

X

j=0

X

16k

j

<k

j−1

<···<k

1

<k

0

(−1)

j

"

_j−1

Y

i=0



2k

i

k

i

− k

i+1

# 

2k

j

k

j



k

j

1 + k

j

= 0.

We ould have expressed formulas (A.1) and (A.2) in terms of Fourier oe ients of

primitive forms to determine the lower order terms. However, this is denitely not the

bestway topro eedsin e it onsistsinde omposingthepolynomial

X

K

− X

K−2

inthe anoni al basis of

Q[T ]

and de omposing again ea h element of this anoni al basisin theChebyshev basis

(X

ℓ

)

ℓ∈N

.

Appendix B. S.J.Miller'sidentity and Cheby hev polynomials

S.J. Miller([10 ,Equation(3.12) Page 6℄) re ently proved that

α

f

(p)

K

+ β

f

(p)

K

=

X

06k6K

k≡K

(mod 2)

c

K,k

λ

f

(p)

k

(B.1)

where

c

K,k

= 0

if

k ≡ K + 1 (mod 2)

and

c

0,0

= 0,

c

2K,0

= 2(−1)

K

(K > 1),

c

2K,2L

=

2(−1)

K+L

_{K(K + L − 1)!}

(2L)!(K − L)!

(1 6 L 6 K),

c

2K+1,2L+1

=

(−1)

K+L

_{(2K + 1)(K + L)!}

(2L + 1)!(K − L)!

(0 6 L 6 K).

We would like to give a qui k proof of this identity, the ru ial tool being Cheby hev

polynomials.

Proof of equation (B.1) . We knowthat

α

f

(p)

K

+ β

f

(p)

K

= X

K

(λ

f

(p)) − X

K−2

(λ

f

(p))

for

K > 2

a ordingto (2.7) . Thus, theproof onsists in de omposing thepolynomial

X

K

− X

K−2

inthe anoni al basisof

Q[T ]

. This an be done via(2.12) . Itentailsthat

α

f

(p)

K

+β

f

(p)

K

=

X

06u6K−2

u≡K

(mod 2)

(−1)

(K−u)/2



(K + u)/2

u



+



(K + u)/2 − 1

u



λ

f

(p)

u

+

X

K−16u6K

u≡K

(mod 2)

(−1)

(K−u)/2



(K + u)/2

u



λ

f

(p)

u

,

whi his anequivalent formulationof (B.1) .



Remark B.1 Equation(B.1) ouldbeusedtore overthelower orderterms omingfrom

P

_q

3

[Φ; r]

but,on e again, it is not the most lever way to pro eed sin e it would imply de omposingthepolynomials

X

K

−X

K−2

inthe anoni albasisof

Q[T ]

atthebeginning ofthepro ess andde omposingthe polynomials

T

j

inthebasis

(X

r

)

r>0

justbeforethe endoftheproofinordertobeabletoapplysometra eformulafortheFourier oe ients

of uspforms.

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L

-fun tionsat

s

= 1

,Int. Math.Res.Not.(2004),no. 31,15611617. MRMR2035301(2005f:11094)

[2℄ StephenGelbartandHervéJa quet,Arelationbetweenautomorphi representationsof

GL(2)

and

GL(3)

,Ann.S i.É oleNorm.Sup.(4)11(1978),no.4,471542. MRMR533066(81e:10025) [3℄ E.He ke,ÜberdieBestimmungDiri hlets herReihendur hihreFunktionalglei hung,Math.Ann.

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L

-fun tions, Inst. HautesÉtudesS i.Publ.Math.(2000),no.91, 55131(2001). MRMR1828743 (2002h:11081) [7℄ Henry H.Kim, Fun toriality forthe exterior square of

GL

4

and thesymmetri fourth of

GL

2

,J.

[8℄ Henry H. Kim and Freydoon Shahidi, Cuspidality of symmetri powers with appli ations, Duke Math.J.112(2002),no.1,177197. MRMR1890650 (2003a:11057)

[9℄ , Fun torialprodu tsfor

GL

2

× GL

3

andthe symmetri ube for

GL

2

, Ann.ofMath. (2) 155 (2002), no. 3, 837893, With an appendix by Colin J. Bushnell and Guy Henniart. MR

MR1923967 (2003m:11075)

[10℄ StevenJ.Miller, Anidentity forsumsofpolylogarithm fun tions.

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l

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L

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