Related papers: L-functions as distributions

We study a new orthogonal family of $L$-functions associated with holomorphic Hecke newforms of level $q$, averaged over $q \asymp Q$. To illustrate our methods, we prove a one level density result for this family with the support of the…

This article extends classical one variable results about Euler products defined by integral valued polynomial or analytic functions to several variables. We show there exists a meromorphic continuation up to a presumed natural boundary,…

We describe the action of the infinite-dimensional Lie algebra $W_{1+\infty}$ and its B-type analogue on Schur and Schur Q-functions, respectively, using formal distributions framework. We observe an interesting self-duality property…

We develop a formal group--theoretic framework for the Riemann zeta function by treating its Euler product as an element of the multiplicative formal group $\widehat{\mathbb{G}}_m$ and its logarithm as the associated formal group logarithm.…

A family of global integrals representing a product of tensor product (partial) $L$-functions: $ L^S(s,\pi\times\tau_1)L^S(s,\pi\times\tau_2)... L^S(s,\pi\times\tau_r) $ are established in this paper, where $\pi$ is an irreducible cuspidal…

Let $\pi$ be a square integrable representation of a classical group and let $\rho$ be a cuspidal representation of a general linear group. We can define in two different ways an L-function $L(\rho \times \pi,s)$: first we can use the…

We present an overview of bounds on zeros of $L$-functions and obtain some improvements under weak conjectures related to the Goldbach problem.

We construct an Euler system in the cohomology of the tensor product of the Galois representations attached to two modular forms, using elements in the higher Chow groups of products of modular curves. We use this Euler system to prove a…

Assuming the generalized Riemann hypothesis, we prove quantitative estimates for the number of simple zeros on the critical line for the L-functions attached to classical holomorphic newforms.

We establish "higher depth" analogues of regularized determinants due to Milnor for zeros of cuspidal automorphic L-functions of GL_d over a general number field. This is a generalization of the result of Deninger about the regularized…

The thesis gave a fine study on the distribution of the coefficients of automorphic L-functions for GL(m) with m>1. In particular we have treated two types of problems: change of signs of these coefficients (when they are real) and their…

We propose a functional integral representation for Archimedean L-factors given by products of Gamma-functions. The corresponding functional integral arises in the description of type A equivariant topological linear sigma model on a disk.…

While several instances of shifted convolution problems for GL(3) x GL(2) have been solved, the case where one factor is the classical divisor function and one factor is a GL(3) Fourier coefficient has remained open. We solve this case in…

We prove a new automorphy lifting theorem for l-adic representations where we impose a new condition at l, which we call `potential diagonalizability'. This result allows for `change of weight' and seems to be substantially more flexible…

We believe that Euler constant is not just the "renormalized" value of the Riemann zeta function in 1. In a sense that we shall clarify it is in fact the normal and natural value of zeta of 1. In this paper we first propose a limit…

We provide estimates for the convolution product of an arbitrary number of "resurgent functions", that is holomorphic germs at the origin of $C$ that admit analytic continuation outside a closed discrete subset of $C$ which is stable under…

Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients lambda_n, indexed by the integers. We define similar coefficients attached to principal automorphic L-functions over GL(N). We…

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 study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number $a\neq0$ and a function from the Selberg class $\mathcal{L}$, we…

We show that a considerable part of the theory of (ultra)distributions and hyperfunctions can be extended to more singular generalized functions, starting from an angular localizability notion introduced previously. Such an extension is…