Estimates for Fourier Coefficients of Cusp Forms in Weight Aspect

Let f be a holomorphic Hecke eigenform of weight k for the modular groupΓ = SL2(Z) and let λf(n) be the n-th normalized Fourier coefficient. In this paper, by a new estimate of the second integral moment of the symmetric square L-function related to f, the estimate 1λf(n21) x2 k2(log(x + k))6n≤x is established, which improves the previous result.     

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．     

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.     

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.

     

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.      

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).     

H^P（S）中函数的Fourier系数

本得到几个关于多变量H^p空间中函数Fourier系数的不等式,与 单变量相应情形相比,我们的证明方法有较大的变化。     

Symmetric Squares of Hecke L-Functions and Fourier Coefficients of Cusp Forms

Let S _k (₀(N)) be the space of cusp forms of even weight k for ₀ (N), let be the set of all newforms in S _k ( ₀ (N)), and let be the symmetric square of the Hecke L-function of a form . It is proved that for N=p we have

Rankin-Selberg without unfolding and bounds for spherical Fourier coefficients of Maass forms

We use the uniqueness of various invariant functionals on irreducible unitary representations of in order to deduce the classical Rankin-Selberg identity for the sum of Fourier coefficients of Maass cusp forms and its new anisotropic analog. We deduce from these formulas non-trivial bounds for the corresponding unipotent and spherical Fourier coefficients of Maass forms. As an application we obtain a subconvexity bound for certain -functions. Our main tool is the notion of a Gelfand pair from representation theory.

     

On Fractional Power Moments of Zeta-Functions Associated with Certain Cusp Forms

We prove upper and lower bounds for fractional moments of zeta-functions attached to certain cusp forms on the critical line; the upper bound being conditional subject to the truth of the Riemann hypothesis.      

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 Whittaker models and the vanishing of Fourier coefficients of cusp forms

The purpose of this paper is to construct examples of automorphic cuspidal representations which possess a ψ-Whittaker model even though their ψ-Fourier coefficients vanish identically. This phenomenon was known to be impossible for the groupGL(n), but in general remained an open problem. Our examples concern the metaplectic group and rely heavily upon J L Waldspurger’s earlier analysis of cusp forms on this group. This research was partially supported by Grant No. 8400139 from the United States-Israel Bi National Science Foundation (BSF), Jerusalem, Israel.     

尖点形式L函数的广义零点密度估计(英文)

In this paper a zero-density estimate of the large sieve type is given for the automorphic L-function L f (s,χ),where f is a holomorphic cusp form and χ a Dirichlet character of mod q.     

带有尖形式Fourier系数的指数和估计

设λ_f/(n)是全模群Γ上权为k的全纯Hecke特征形f的第n个Fourier系数,Λ(n)是Mangoldt函数.本文得到了如下估计∑_(Xn≤2X)Λ(n)λ_f(n)e(n~(1/2)α)■f,αX~(5/6)(logX)~(13/2),(α0),改进了Zhao的结果。     

Estimates for Sums of Coefficients of Dirichlet Series with Functional Equation

Suppose we have a Dirichlet series L(s) = _{n = 1} a _n n ^–s such that it, and its twists by Dirichlet characters have analytic continuation and a functional equation of a specific kind. Suppose also that the root numbers of the twists are equidistributed on the unit circle. The purpose of this note is to get an estimate for the quantity for a prime modulus p.We use a modification of the method of Chandrasekharan and Narasimhan and we use in an essential way a Rankin-Selberg type estimate for the average of |a _n|².     

On the Poles of Rankin-Selberg Convolutions of Modular Forms

The Rankin-Selberg convolution is usually normalized by the multiplication of a zeta factor. One naturally expects that the non-normalized convolution will have poles where the zeta factor has zeros, and that these poles will have the same order as the zeros of the zeta factor. However, this will only happen if the normalized convolution does not vanish at the zeros of the zeta factor. In this paper, we prove that given any point inside the critical strip, which is not equal to and is not a zero of the Riemann zeta function, there exist infinitely many cusp forms whose normalized convolutions do not vanish at that point.

     

On the Fourth Moment of Coefficients of Symmetric Square L-Function

Let f(z) be a holomorphic Hecke eigencuspform of weight k for the full modular group. Let ?? _f (n) be the nth normalized Fourier coefficient of f(z). Suppose that L(sym² f, s) is the symmetric square L-function associated with f(z), and $ \lambda _{sym^2 f} (n) $ (n) denotes the nth coefficient L(sym² f, s). In this paper, it is proved that $$ \sum\limits_{n \leqslant x} {\lambda _{sym^2 f}^4 (n)} = xP2(\log x) + O(x^{\frac{{79}} {{81}} + \varepsilon } ), $$ , where P ₂(t) is a polynomial in t of degree 2. Similarly, it is obtained that $$ \sum\limits_{n \leqslant x} {\lambda _f^4 (n^2 )} = x\tilde P2(\log x) + O(x^{\frac{{79}} {{81}} + \varepsilon } ), $$ , where $ \tilde P_2 (t) $ is a polynomial in t of degree 2.     

Estimates of Fourier Coefficients of Siegel Cusp Forms for Subgroups and in the Case of Small Weight

Here we develop estimates for Fourier coefficients of Siegelcusp forms. First we consider the case of Siegel modular formsfor the full modular group