Related papers: Analytic $L$-functions: Definitions, Theorems, and…

A well-known principle states that a congruence between objects should give rise to a corresponding congruence between the special values of $L$-functions attached to these objects. We computationally investigate this principle for…

In this paper we define a function $L$ which will allow us to (separately or simultaneously) generalize many theorems from Number Theory obtained by Wilson, Fermat, Euler, Gauss, Lagrange, Leibniz, Moser, and Sierpinski.

We propose a relationship between the cohomology of arithmetic groups, and the motivic cohomology of certain (Langlands-)attached motives. The motivic cohomology group in question is that related, by Beilinson's conjecture, to the adjoint…

Many important analytic statements about automorphic forms, such as the analytic continuation of certain L-functions, rely on the well-known rapid decay of K-finite cusp forms on Siegel sets. We extend this here to prove a more general…

$L$-functions typically encode interesting information about mathematical objects. This paper reports 29 identities between such functions that hitherto never appeared in the literature. Of these we have a complete proof for 9; all others…

This is an expanded version of the author's lecture at the XX Congresso U.M.I., held in Siena in September 2015. After a brief review of L-functions, we turn to the classical converse theorems of H.Hamburger, E.Hecke and A.Weil, and to some…

We consider a general form of L-function L(s) defined by an Euler product and satisfies several analytic assumptions. We show several asymptotic formulas for L(1) and log L(1). In particular those asymptotic formulas are valid for Dirichlet…

The geometric Langlands correspondence for complex algebraic curves differs from the original Langlands correspondence for number fields in that it is formulated in terms of sheaves rather than functions (in the intermediate case of curves…

We unconditionally construct cyclotomic p-adic L-functions for Rankin-Selberg convolutions for GL(n+1) x GL(n) over arbitrary number fields, and show that they satisfy an expected functional equation.

The L-function $ L(\rho_\lambda, s) $ of an almost everywhere unramified $ \lambda $-adic representation $ \rho_\lambda $ of a global function field $ \mathbb{F}_q(C) $ is known to be a rational function in $ q^{-s} $ satisfying a…

In this paper, we have proved Selberg's Central Limit Theorem for $GL(3)$ $L$-functions associated with the Hecke-Maass cusp form $f$. Moreover, we have proved the independence of the automorphic $L$-functions.

We define L-functions for the class of real-analytic modular forms recently introduced by F. Brown. We establish their main properties and construct the analogue of period polynomial in cases of special interest, including those of modular…

Assuming the Generalized Riemann Hypothesis, we provide uniform upper and lower bounds with explicit main terms for $\log{\left|\cL(s)\right|}$ for $\sigma \in (1/2,1)$ and for functions in the Selberg class. In particular, we focus on the…

We prove the equivalence of two conjectural constructions of unramified cuspidal automorphic functions on the adelic group GL_n(A) associated to an irreducible l-adic local system of rank n on an algebraic curve X over a finite field. The…

We give a new heuristic for all of the main terms in the integral moments of various families of primitive L-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical…

We present a pairing of automorphic distributions that applies in situations where a Lie group acts with an open orbit on a product of generalized flag varieties. The pairing gives meaning to an integral of products of automorphic…

We propose the Lie-algebraic interpretation of poly-analytic functions in $L_2(\C,d\mu)$, with the Gaussian measure $d\mu$, based on a flag structure formed by the representation spaces of the $\mathfrak{sl}(2)$-algebra realized by…

We prove three main results: all Langlands-Shahidi automorphic $L$-functions over function fields are rational; after twists by highly ramified characters our automorphic $L$-functions become polynomials; and, if $\pi$ is a globally generic…

We study a Lax pair in a $2$-parameter Lie algebra in various representations. The overlap coefficients of the eigenfunctions of $L$ and the standard basis are given in terms of orthogonal polynomials and orthogonal functions. Moreover,…

We study the behaviour of automorphic L-Invariants associated to cuspidal representations of GL(2) of cohomological weight 0 under abelian base change and Jacquet-Langlands lifts to totally definite quaternion algebras. Under a standard…