Estimates for Fourier Coefficients of Cusp Forms in Weight Aspect∗

Hengcai TANG

1 Introduction

Let Hk(Γ)be the space of Hecke-eigen cusp forms of even integral weight k for Γ =SL(2,Z).Suppose that f(z)has the following Fourier expansion at the cusp∞:

where e(x):=e2πixand the n-th normalized Fourier coefficient λf(n)of f is the eigenvalue under the Hecke operator Tn.Then from the theory of Hecke operators,the following is nowadays widely known:

(i)λf(n)is real and satisfies the multiplicative property

for all integers m≥1 and n≥1.

(ii)For all n≥1,

where d(n)is the divisor function.This is the well-known Petersson-Ramanujan conjecture which was proved by Deligne[2]in 1974.As a corollary,he proved that for any ε ＞ 0,

Later,many authors considered the summation and the Sato-Tate conjecture implies that

with ρ ≈ 0.151.

Let

It was Ivi´c[7]who first considered oscillations of the Fourier coefficients over squares.Based on the prime number theorem,he successfully showed that

In 2006,Fomenko[4]improved Ivi´c’s result by proving

In 2006,Sankaranarayanan[13]proved that

uniformly for k≪where the implied constant is absolute.Later,L¨u[10]showed that,in fact,for k≥2,

where the implied constant is absolute.Ichihara[6]obtained the best upper bound for x which states that

where the implied constant is effective.

The purpose of this paper is to improve the above results in the weight aspect.By a new bound for the second integral moment of the symmetric square L-function L(s,sym2f)at the critical line(see Proposition 2.1 in the next section),we get the following result.

Theorem 1.1 Let f be a holomorphic Hecke eigenform of weight k for Γ and let λf(n)be the n-th normalized Fourier coefficient.Then we have

where the implied constant is absolute and does not depend on f.

2 Preliminaries

The behavior of S(x)is intimately connected with the symmetric square L-function associated with f which is defined by

where ℜs ＞ 1 and ζ(s)is the Riemann zeta function.For convenience,hereafter,we write F=sym2f which is a cuspidal automorphic form for SL(3,Z)by the Gelbart-Jacquet lift(see[5]).The functional equation of L(s,F)is given by

where

is the Archimedean local factor.It is known that Λ(s,F)can be extended to an entire function and satisfies(see[8])

Denote by λF(n)the n-th coefficient of the Dirichlet series expansion of L(s,F).This means that for ℜs ＞ 1,

From(2.1),we have

Comparing with another special GL(3)L-function ζ3(s),

we have,by(1.2),

By Mbius inversion and(2.3),we have

where μ(m)is the Mbius function.As in Ichihara[6],we transform the question of estimating S(x)into studying the sumin the following way:

Then Theorem 1.1 follows from the following result.

Proposition 2.1 Let f be a holomorphic Hecke eigenform of weight k for Γ and L(s,F)the symmetric square L-function associated with f.Denote by λF(n)the n-th normalized Fourier coefficient of L(s,F).Then we have

where the implied constant is effective and does not depend on F.

To prove Proposition 2.1,we need the following four lemmas.The first one is related to the uniform convexity bound for L(s,F).In order to give a new estimate for the mean square of L(s,F)at the critical line,we introduce the approximate functional equation of L(s,F)and a classical result due to Montgomery and Vaughan[11].The difficulty is that the weight aspect should be considered.

Lemma 2.1 Let τ=(|t|+1)(k+|t|)2.Then

holds for

Proof By(2.4),we have

On the other hand,by the functional equation in(2.2),we have

where

In[13],Sankaranarayanan proved that for any∈＞0 and−1+∈≤ℜs=c＜0,

Then we have

It follows that

Replacing the formulas(3.4.1)and(3.4.2)in the paper of Sankaranarayanan[13]by(2.6)–(2.7),we complete the proof.

Lemma 2.2Then for any Y ≥2,we have

Proof By applying Mellin’s inversion formula to Γ(z),we have

where x ＞ 0 and(a)means the line ℜz=a.and sum over n on the both sides.Finally we get

where we have usedMoving the line of integration towe have

where we have used Lemma 2.1 in the last estimate.Following from(2.8)–(2.9),we complete the proof of this lemma.

Lemma 2.3be a set of arbitrarily complex numbers.Then

The above formula also remains valid if N=∞,provided that the series on the right-hand side converge.Furthermore,

provided that the summations in the formula converge.

Proof See Theorem 5.2 of Ivi´c[7].

Lemma 2.4 Let s=+it,T≤t≤2T.Then for sufficiently large T＞2,we have

Proof The approximate functional equation of L(s,F)states that

where s=+it,T≤t≤2T and Y≥2.Hence it is sufficient to prove

and

Taking Y=(T+k)and using Lemma 2.3,we have

where we have used the partial summation formula and the estimate(see[15])

Hence the estimate(2.10)follows.Trivially,we also have I3≪(T+k)(log(T+k))6because of the choice of Y.Thus it only remains to prove(2.11).

By the functional equation of L(s,F),we obtain

Next,we split L(1−s−z,F)into two parts.Then

By the Cauchy’s inequality and Lemma 2.3,we obtain

Here we have used the partial summation and Y=T12(T+k).

For I21,it is slightly different from the estimation of I22.Moving the inner integration to the parallel segment with ℜz=we have

Next,following the step of the evaluation of I22,we get

This completes the proof.

Remark 2.1 Sankaranarayanan[13]pointed out that mean value theorems play an important role in L-function theory and he established the following result:

holds for sufficiently large T.By the observation of Γ-functions,we obtainedk instead ofwhich implies the convexity bound in the k-aspect,i.e.,If one can reduce the power of T,the subconvexity bound of L(s,F)in the t-aspect will be given obviously.Another way is to evaluate the integral in short intervals.Recently,Li[9]proved that

forwhich implies thatUnfortunately,the subconvexity bound in k-aspect is rarely given.There has been no other result up to now,except one for the weak subconvexity of Soundararajan[14].

3 Proof of Proposition 2.1

Without loss of generality,we assume that x is not an integer.By Perron’s formula(see Davenport[1,p.105]),we have

whereand T≤x is a parameter to be chosen later.Following from the argument of Ramachandra and Sankaranarayanan[12],we have

where∈＞ 0 can be arbitrarily small.Takingand moving the line of integration in(3.1)toby the residue theorem,we get

For the integral in the O-term,we have

where we have used Lemma 2.1.

Next,we put T0=max{e30,8k}and split the first integral in(3.1)into three pieces,i.e.,

By Cauchy’s inequality and Lemma 2.4,we have

For the other two integrals,we follow the argument of Ichihara[6].Consider

where the integral interval L means two segments which satisfy σ =and T0≤ |t|≤ T.Divide the interval L into Lj(0≤j≤J)with J satisfyingdenotes the intervalandThe argument of the first case implies that

Furthermore,Section 4 of Ichihara[6]gives the bound of the integral over Lj(0≤j≤J−1),i.e.,

The only difference is that we have used(2.12)instead of the estimate

Obviously,J≪logx.Combining the above estimates,we finally get

This proves Proposition 2.1.

Acknowledgements This work was completed when the author visited Institut´Elie Cartan de Lorraine supported by Henan University.The author would like to thank Professor Wu Jie for his encouragement.The author is grateful to the anonymous referee for his or her comments,especially for his or her suggestions on how to improve the result of Proposition 2.1.

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