We have ${\sum }_{K-G\le k_j\le K+G}{\alpha }_j{H}_j^3\left(\frac{1}{2}\right){\ll }_{ϵ}G{K}^{1+ϵ}$ for $K^{ϵ}\le G\le K,\text{where}{\alpha }_j={\left|{\rho }_j\left(1\right)\right|}^2{(cosh\pi k_j)}^{-1},\text{and}{\rho }_j\left(1\right)$ is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue ${\lambda }_j=k_j^2+\frac{1}{4}$ to which the Hecke series $H_j\left(s\right)$ is attached. This result yields the new bound $H_j(\frac{1}{2}{\ll }_{ϵ}{k}_j^{\frac{1}{3}+ϵ}.$

Let ${\lambda }_f\left(n\right)$ be the nth normalized Fourier coefficient of a holomorphic or Maass cusp form f for SL(2,ℤ). We establish the asymptotic formula for the summatory function ${\sum }_{\begin{array}{c}n\le x\\ n\equiv l\left(modq\right)\end{array}}{\left|{\lambda }_f\left(n\right)\right|}^{2j}$ as x → ∞, where q grows with x in a definite way and j = 2,3,4.

Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma =\mathrm{SL}(2,\mathbb{Z})$. Denote by ${\lambda }_f\left(n\right)$ the $n$th normalized Fourier coefficient of $f$. We are interested in the average behaviour of the sum $$\sum _{a^2+b^2\le x}{\lambda }_f^j(a^2+b^2)$$ for $x\ge 1$, where $a,b\in \mathbb{Z}$ and $j\ge 9$ is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power $L$-functions and Rankin-Selberg $L$-functions.

In the paper the asymptotics for Dirichlet polynomials associated to certain cusp forms are obtained.

A formula for the mean value of multiplicative functions associated to certain cusp forms is obtained. The paper is a continuation of [4].

In this paper, we are interested in exploring the cancellation of Hecke eigenvalues twisted with an exponential sums whose amplitude is √n at prime arguments.

Considérons les formes linéaires $f\mapsto L(f,\chi ,1)$ sur l’espace vectoriel des formes paraboliques de poids $2$ pour le groupe de congruence ${\Gamma }_0\left(p\right)$, avec $\chi$ un caractère de Dirichlet modulo $p$. Par le produit scalaire de Petersson, on peut leur associer des formes paraboliques. Cet article détermine le produit scalaire de deux de ces formes, pour deux caractères de Dirichlet non triviaux de parités différentes.