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.     

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     

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.      

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 relations among best approximation and Fourier coefficients

We discuss the relations among the best approximation E _n (f) and the Fourier coefficients of a function, under the conditions that and , where MVBVS^* is the class of the so-called Strong Mean Value Bounded Variation Sequences. This work was supported by NSERC RCD grant and AARMS of Canada (Yu D S), by NSERC of Canada (Zhou P), and by the National Natural Science Foundation of China (Grant No. 10471130) and the Open Fund (Grant No. PLN0613) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University) (Zhou S P)     

Quadratic forms connected with Fourier coefficients of Maass cusp forms

For the normalized Fourier coefficients of Maass cusp forms λ(n) and the normalized Fourier coefficients of holomorphic cusp forms a(n), we give the bound of ∑m12+m22+m32≤xλ(m12+m22+m32)Λ(m12+m22+m32) and ∑m12+m22+m32≤xa(m12+m22+m32)Λ(m12+m22+m32).     

关于立方幂补数倒数的1/3次均值

利用初等方法得到了立方幂补数S(n)倒数的1/3次均值的一个渐近公式,补充了有关文献的结论.     

Sign changes of Fourier coefficients of Maass eigenforms

Let f be a Maass eigenform that is a new form of level N on Γ0(N),with Laplace eigenvalue 1/4 + νf2 . Then,all its Fourier coefficients {λf(n)}∞n=1 are real,and we may normalize so that λf (1) = 1. It is proved,in this paper,that the first sign change in the sequence {λf(n)}n∞=1 occurs at some n satisfying n《{(3+|νf|2)N }1/2-δ and (n,N)=1. This generalizes the previous results for holomorphic Hecke eigenforms.     

On the even power mean of a sum analogous to Dedekind sums

Summary Our purpose is to extend results due to P. Chandra and L. Leindler concerning the order of approximation by means of Fourier series for functions belonging to generalized Lipschitz-classes.     

关于三次方幂补数的均值

设n是正整数，s(n)为n的三次方幂补数．本文的主要目的是研究s(n)的均值分布性质，解决文[2]中F．Smarandache教授提出的第28个问题，给出了一个有趣的渐近公式。     

关于四次及六次混合指数和的均值

本文的主要目的是运用初等及代数方法研究四次及六次混合指数和的均值计算问题,并给出几个有趣的计算公式和转换公式.     

On the fourth moment of holomorphic Hecke cusp forms

We find the average value of the fourth moment of holomorphic Hecke cusp forms of weight within a dyadic interval, up to an error which saves a power of the weight.     

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 number of Fourier coefficients that determine a Hilbert modular form

We estimate the number of Fourier coefficients that determine a Hilbert modular cusp form of arbitrary weight and level. The method is spectral (Rayleigh quotient) and avoids the use of the maximum principle.

     

关于原数列及其均值

设p为素数,n为正整数,Sp(n)是其阶乘能被pn整除的最小正整数.本文研究了数列Sp(n)的均值性质,并给出了一个较强的渐进公式.     

Smarandache幂函数的均值

对于给定的自然数n,Smarandache幂函数SP(n)定义为SP(n)=min{m: n|mm,m∈N}．本文研究了这个函数的均值分布性质,并利用解析方法得到了Smarandache幂函数的一个较强的均值公式．     

关于三次高斯和的均值

主要研究三次高斯和的均值性质，并给出一个较强的渐近公式。     

On the general Kloosterman sum and its fourth power mean

The main purpose of this paper is to study the asymptotic property of the fourth power mean of the general Kloosterman sums, and give an interesting calculating formula.     

Dimension formula of cusp form spaces on half-spaces of quaternions

A dimension formula for the spaces of cusp forms defined on quatemionic half-spacea of degree two is obtained by Selberg trace formula, and the contributions of some conjugacy classes are calculated. Some results on the classification of the conjugacy classes of modular group are obtained.     

关于Hurwitz Zeta-函数导数的一类新型均值公式

本文利用解析方法给出(?)(s,α)对参数α的二次积分均值的一个很强的渐近公式．