Related papers: Simultaneous non vanishing of $GL(3)$ $L$-function…

We show that for at least $\frac{5}{13}$ of the primitive Dirichlet characters $\chi$ of large prime modulus, the central value $L(\frac{1}{2},\chi)$ does not vanish, improving on the previous best known result of $\frac{3}{8}$.

We present some monotonicity results for Dirichlet $L$-functions associated to real primitive characters. We show in particular that these Dirichlet $L$-functions are far from being logarithmically completely monotonic. Also, we show that,…

We evaluate asymptotically the smoothed first moment of central values of families of quadratic, cubic, quartic and sextic Hecke $L$-functions over various imaginary quadratic number fields of class number one, using the method of double…

A quadratic twist of the L-function associated with a modular form is known to satisfy a functional equation, which may be even or odd. A result due to Gross and Zagier explicitly computes the central value of the L-function or its…

We establish upper bounds for shifted moments of cubic and quartic Dirichlet $L$-functions under the generalized Riemann hypothesis. As an application, we prove bounds for moments of cubic and quartic Dirichlet character sums.

We prove an asymptotic formula for the second moment of automorphic L-functions of even weight and prime power level. The error term is estimated uniformly in all parameters: level, weight, shift and twist.

We compute the expected value of Dirichlet $L$-functions defined over $\mathbb{F}_q[T]$ attached to cubic characters evaluated at an arbitrary $s \in (0,1)$. We find a transition term at the point $s=\frac{1}{3}$, reminiscent of the…

We prove that a cuspidal automorphic representation of GL(3) over any number field is determined by the quadratic twists of its central value. In the case of a non-Gelbart-Jacquet lift, the result is conditional on the analytic behavior of…

We establish lower bounds for the $2k$-th moment of families of quadratic Dirichlet $L$-functions at the central point for all real $k<0$, assuming a conjecture of S. Chowla on the non-vanishing of these $L$-values.

We prove an asymptotic formula for the second moment of central values of Dirichlet $L$-functions restricted to a coset. More specifically, consider a coset of the subgroup of characters modulo $d$ inside the full group of characters modulo…

We consider the multiple Dirichlet series associated to the $k$th moment of real Dirichlet $L$-functions, and prove that it has a meromorphic continuation to a specific region in $\mathbb{C}^{k+1}$, which is conditional under the…

This paper is second in a series of three papers; the first of which is "Summation Formulas, from Poisson and Voronoi to the Present" (math.NT/0304187), and the third of which is "Automorphic Distributions, L-functions, and Voronoi…

We explicitly evaluate a special type of multiple Dirichlet $L$-values at positive integers in two different ways: One approach involves using symmetric functions, while the other involves using a generating function of the values. Equating…

We calculate certain "wide moments" of central values of Rankin--Selberg $L$-functions $L(\pi\otimes \Omega, 1/2)$ where $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_2$ over $\mathbb{Q}$ and $\Omega$ is a Hecke character…

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…

In a previous article we had proved an algebraicity result for the central critical value for L-functions for GL(n) x GL(n-1) over Q assuming the validity of a nonvanishing hypothesis involving archimedean integrals. The purpose of this…

We prove an asymptotic formula with a power-saving error term for a specific weighted second moment of $\mathrm{GL}(2)\times \mathrm{GL}(2)$ Rankin-Selberg $L$-function, $L(1/2,\pi\otimes \pi_0)$ over any number field $F$ where $\pi$ runs…

In the first chapter, we will present a computation of the square value of the module of L functions associated to a Dirichlet character. This computation suggests to ask if a certain ring of arithmetic multiplicative functions exists and…

We prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment of central values of L-functions of any two (possibly equal) fixed cusp forms f, g twisted by all primitive characters modulo q, valid for…

We obtain nonvanishing estimates for central values of certain self-dual Rankin-Selberg $L$-functions on $\operatorname{GL}_2({\bf{A}}_F) \times \operatorname{GL}_2({\bf{A}}_F)$, and more generally $\operatorname{GL}_r({\bf{A}}_F) \times…