Related papers: Weighted average values of automorphic $L$-functio…

We evaluate the smoothed first moment of central values of a family of qudratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series. The asymptotic formula we obtain has an error term of size…

For $n\in\mathbb{N}$ and $\ell\in\{0,1,\dots,n\}$, we consider the function extracting the $\ell$th coefficient of the Ehrhart polynomials of lattice polytopes in $\mathbb{R}^n$. These functions form a basis of the space of unimodular…

We prove, in standard notation from spectral theory, the asymptotic formula ($B>0$) $$ \sum_{\kappa_j\le T}\alpha_j H_j(1/2) = ({T\over\pi})^2 - BT\log T + O(T(\log T)^{1/2}), $$ by using an approximate functional equation for $H_j(1/2)$…

Let $p\geq 3$ be a prime number and $K$ be a quadratic imaginary field in which $p$ splits as $\mathfrak{p}\overline{\mathfrak{p}}$. Let $\mathcal{F}$ be a cuspidal Bianchi eigenform over $K$ of weight $(k,k)$, where $k\geq 0$ is an…

The summatory function of $t_j(n^2)$ is estimated, where $H_j(s) = \sum_{n=1}^\infty t_j(n)n^{-s}$ is the Hecke series of a non-holomorphic cusp form. The analogous problem of holomorphic cusp forms is also treated.

Let f be a classical holomorphic cusp form for SL_2(Z) of weight k which is a normalized eigenfunction for the Hecke algebra, and let \lambda(n) be its eigenvalues. In this paper we study "shifted convolution sums" of the eigenvalues…

We establish asymptotic formulas for sums of reciprocals of primes in arithmetic progressions, generalizing recent results on multiple Mertens evaluations by Tenenbaum, Qi, and Hu. Specifically, for any fixed constant $K>0$, we derive…

We study the Lambert series $\mathscr{L}_q(s,x) = \sum_{k=1}^\infty k^s q^{k x}/(1-q^k)$, for all $s \in \mathbb{C}$. We obtain the complete asymptotic expansion of $\mathscr{L}_q(s,x)$ near $q=1$. Our analysis of the Lambert series yields…

Let $f$ be a primitive form of weight $2k+j-2$ for $SL_2(Z)$, and let $\mathfrak p$ be a prime ideal of the Hecke field of $f$. We denote by $SP_m(Z)$ the Siegel modular group of degree $m$. Suppose that $k \equiv 0 \mod 2, \ j \equiv 0…

We study the average of the product of the central values of two $L$-functions of modular forms $f$ and $g$ twisted by Dirichlet characters to a large prime modulus $q$. As our principal tools, we use spectral theory to develop bounds on…

In this note, we give a detailed proof of an asymptotic for averages of coefficients of a class of degree three $L$-functions which can be factorized as a product of a degree one and a degree two $L$-functions. We emphasize that we can…

Let $f(z)=q+\sum_{n\geq 2}a(n)q^n$ be a weight $k$ normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in \cite{AH} for $k=2$ by ruling out or locating all…

We prove an asymptotic formula for $F\left(1/4-it+ir,1/4-it-ir,1/2;x \right)$ as $r, t\to\infty$ and $\alpha=r/t\to0.$ This special case of the Gauss hypergeometric function appears in the explicit formula for the first moment of Maass form…

In this article, we establish an average behaviour of the normalised Fourier coefficients of the Hecke eigenforms supported at the integers represented by any primitive integral positive definite binary quadratic form of fixed discriminant…

Let $G(g;x):=\sum_{n\leq x}g(n)$ be the summatory function of an arithmetical function $g(n)$. In this paper, we prove that we can write weighted averages of an arbitrary fixed number $N$ of arithmetical functions $g_{j}(n),\,j\in\left\{…

We deduce an asymptotic formula with error term for the sum $\sum_{n_1,\ldots,n_k \le x} f([n_1,\ldots, n_k])$, where $[n_1,\ldots, n_k]$ stands for the least common multiple of the positive integers $n_1,\ldots, n_k$ ($k\ge 2$) and $f$…

In this article we present a heuristic model that describes the asymptotic behaviour of the number of primes p such that the p-th coefficient of a given eigenform is a rational integer. We treat the case of a weight 2 eigenform with…

Let $p$ be an odd prime, $q=p^e$, $e\ge 1$, and $\mathbb{F} = \mathbb{F_q}$ denote the finite field of $q$ elements. Let $f: \mathbb{F}^2\to \mathbb{F}$ and $g: \mathbb{F}^3\to \mathbb{F}$ be functions, and let $P$ and $L$ be two copies of…

In this paper we obtain a weighted average formula for special values of $L$-functions attached to normalized elliptic modular forms of weight $k$ and full level. These results are obtained by studying the pullback of a Siegel Eisenstein…

Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a cuspidal automorphic representation $\pi$ of $D_{\mathbb{A}}^\times$ with trivial central character and a cusp form $\phi$ in $\pi$. Using the prehomogeneous zeta…