Average behavior of Fourier coefficients of cusp forms over sum of two squares

Let ${\lambda }_f(n)$ denote the nth normalized Fourier coefficient of the classical holomorphic cusp form of even integral weight $k\ge 2$ for the full modular group $\mathit{SL}(2,\mathbb{Z})$. In this paper, we investigate the average behavior of the power sum$\sum _{a^2+b^2\le x}{\lambda }_f{(a^2+b^2)}^{?}$ for $x\ge 1$, $2\le ?\le 8$ and $a,b,?\in \mathbb{Z}$.     

On the gaps in the Fourier expansion of cusp forms

In this paper we obtain some results on the gap function which measures the size of gaps in the Fourier expansion of cusp forms that are not linear combinations of forms with complex multiplication. We also investigate the nonvanishing of Fourier coefficients of such cusp forms along rational multiples of linear forms in two variables.      

On the Signs of Fourier Coefficients of Cusp Forms

Let be a discrete subgroup of SL(2, ) with a fundamental region of finite hyperbolic volume. (Then, is a finitely generated Fuchsian group of the first kind.) Let 0}} {a(n)e^{2\pi i(n + {\kappa })z/{\lambda }} } ,{ }z \in \mathcal{H}.$$ " align="middle" border="0"> be a nontrivial cusp form, with multiplier system, with respect to . Responding to a question of Geoffrey Mason, the authors present simple proofs of the following two results, under natural restrictions upon . Theorem. If the coefficients a(n) are real for all n, then the sequence {a(n)} has infinitely many changes of sign. Theorem. Either the sequence {Re a(n)} has infinitely many sign changes or Re a(n) = 0 for all n. The same holds for the sequence {Im a(n)}.     

Revisiting Rademacher's Formula for the Partition Function p(n

We provide a new proof of Rademacher's celebrated exact formula for the partition function. Along the way we present a simple treatment of an integral which is ubiquitous in the theory of nonanalytic automorphic forms.     

半整权模形式Fonrier系数的两个注记

对于ｆ（ｚ）＝∑α＿ｎｅ（ｎｚ）∈Ｓ＿ｋ（г＿０（Ｎ））,Ｈ．Ｉｗａｎｉｅｃ￣［２］证明了,其中ｎ为无平方因子正整数．在本文中我们将推广这个结果．     

On the third and fourth power moments of Fourier coefficients of cusp forms

The asymptotic formulae for the third and fourth power moments of Fourier coefficients of cusp forms are proved in this paper.     

A note on Siegel modular forms

This paper gives a new identification for Siegel modular forms with respect to any congruence subgroup by investigating the properties of their Fourier-Jacobi expansions, and verifies a comparison theorem for the dimensions of the spaces Skn (Γn) and J0k, 1 (Γn) with small weight k. These results can be used to estimate the dimension of the space of modular forms.     

The cancellation of Fourier coefficients of cusp forms over different sparse sequences

Let λ f(n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z)∈Sk(Γ).In this paper,we established nontrivial estimates for ∑n≤xλf(ni)λf(nj),where 1 ≤ i j ≤ 4.     

Hecke特征值相邻极大值的均值估计

取f为权为偶数k的全模群Γ = SL(2,Z)的Hecke特征型.定义Asymmf(n)为与f关联的m阶对称幂次L-函数的Dirichlet展开式的第n个正规化系数.本文中,我们给出了Σn0为一个正数.     

A Limit Theorem for the Arguments of Zeta-Functions of Certain Cusp Forms

In this paper, we obtain a limit theorem for the moduli of the arguments of zeta-functions on the critical line of normalized eigenforms.     

On the traces of Hecke operators

From Eichler's results on traces of Hecke operators and making essential use of the theory of Atkin-Lehner on newforms, we derive in a simple manner an equality between the traces of Hecke operators acting on different spaces of cusp forms.     

On Fourier coefficients of Maass cusp forms in 3-dimensional hyperbolic space

In this article we establish the analogue of a theorem of Kuznetsov (theorem 6 of [3]) in the case of 3-dimensional hyperbolic space. We also consider a generalization of this result for higher dimensional hyperbolic spaces and discuss the relevant ingredients of a proof. Dedicated to the memory of Professor K G Ramanathan     

On the distribution and moments of the fourier coefficients of cusp forms

Two theorems about the distribution and moments of the Fourier coefficients of cusp forms are proved in this paper. Project supported by the National Natural Science Foundation of China     

Existence and Weyl’s law for spherical cusp forms

Let G be a split adjoint semisimple group over and a maximal compact subgroup. We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of . This proves a conjecture of Sarnak for -split groups, previously known only for the case G = PGL(n). The key idea amounts to a new type of simple trace formula. Received: April 2005 Revision: June 2006 Accepted: October 2006