Related papers: A Note on the Mean Value of $L$--functions in Func…

Let $A(1,m)$ be the Fourier coefficients of a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form $\pi_1$ and $\lambda(m)$ be those of a $SL(2,\mathbb{Z})$ Hecke holomorphic or Hecke-Mass cusp form $\pi_2$. Let $H\subset[\![…

We give a lower bound for the maximum value of class group $L$-functions attached to $\mathbb{Q}(\sqrt{-D})$ at the central point and show that this value is on average at least $$\exp\Bigg(\delta\sqrt{\frac{\log D \log \log \log D}{\log…

In this paper various analytic techniques are com- bined in order to study the average of a product of a Hecke L- function and a symmetric square L-function at the central point in the weight aspect. The evaluation of the second main term…

For non-negative integers $l_{1}, l_{2},\ldots, l_{n}$, we define character sums $\varphi_{(l_{1}, l_{2},\ldots, l_{n})}$ and $\psi_{(l_{1}, l_{2},\ldots, l_{n})}$ over a finite field which are generalizations of Jacobsthal and modified…

We evaluate the asymptotic size of various sums of G\'al type, in particular $$S( \mathcal{M}):=\sum_{m,n\in\mathcal{M}} \sqrt{(m,n) \over [m,n]},$$ where $\mathcal{M}$ is a finite set of integers. Elaborating on methods recently developed…

Let $R$ be the ring of integers in a finite extension $K$ of $\mathbb{Q}_p$, let $k$ be its residue field and let $\chi:\pi_1(X)\to R^{\times}=GL_{1}(R)$ be a "geometric" rank one representation of the arithmetic fundamental group of a…

Let $\lambda_{\phi}(n)$ be the Fourier coefficients of a Hecke holomorphic or Hecke--Maass cusp form on ${\rm SL}_2(\mathbb Z)$, and $f$ be any multiplicative function that satisfies two mild hypotheses. We establish a non-trivial upper…

Fix a Dirichlet character $\chi$ and a cuspidal GL$(2)$ eigenform $\phi$ with relatively prime conductors. Then we show that there are infinitely many cusp forms $\pi$ on GL$(3)$ such that $L(1/2, \pi \times \chi)$ and $L(1/2, \pi \times…

We study a certain class $\mathcal{P}$ of positive linear functionals $\varphi$ on $L^{\infty}([1,\infty))$ for which $\varphi(f) = \alpha$ if $\lim_{x \to \infty} \frac{1}{x} \int_1^x f(t)dt = \alpha$. It turns out that translations $f(x)…

Using estimates on Hooley's $\Delta$-function and a short interval version of the celebrated Dirichlet hyperbola principle, we derive an asymptotic formula for a class of arithmetic functions over short segments. Numerous examples are also…

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(X_1,X_2)\in\mathbb{Z}[X_1,X_2] $ be an irreducible binary form of degree $3$ and $h$ an arithmetic function. We give some estimates for the average order $\sum_{\substack{|n_1|\leq x,|n_2|\leq x}}h(F(n_1,n_2))$ when $h$ satisfy…

To date, the best methods for estimating the growth of mean values of arithmetic functions rely on the Vorono\"{\i} summation formula. By noticing a general pattern in the proof of his summation formula, Vorono\"{\i} postulated that…

Given a suitable arithmetic function h, we investigate the average order of h as it ranges over the values taken by an integral binary form F. A general upper bound is obtained for this quantity, in which the dependence upon the…

Let $\chi$ denote a primitive, non-quadratic Dirichlet character with conductor $q$, and let $L(s, \chi)$ denote its associated Dirichlet $L$-function. We show that $|L(1, \chi)| \geq 1/(9.12255 \log(q/\pi))$ for sufficiently large $q$, and…

We show that the central value of class group L-functions of CM fields can be expressed in terms of derivatives of real-analytic Hilbert Eisenstein series at CM points. Then, following an idea of Iwaniec and Kowalski we obtain a conditional…

For $m\ge 2$, let $\pi$ be an irreducible cuspidal automorphic representation of $GL_m(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. Let $a_\pi(n)$ be the $n^{th}$ coefficient of the $L$-function attached to $\pi$. Goldfeld and…

An explicit formula for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the odd Dirichlet characters modulo $f>2$ is known. Here we present a situation where we could prove an explicit formula for the…

This paper provides a mean value theorem for arithmetic functions $f$ defined by $$f(n)=\prod_{d|n}g(d),$$ where $g$ is an arithmetic function taking values in $(0, 1]$ and satisfying some generic conditions. As an application of our main…

We use multiple zeta functions to prove, under suitable assumptions, precise asymptotic formulas for the averages of multivariable multiplicative functions. As applications, we prove some conjectures on the average number of cyclic…