A Theorem for Distinct Zeros of L-Functions

TL;DR

This paper provides a simple criterion to determine when two L-functions with functional equations have infinitely many distinct zeros, broadening understanding without relying on advanced analytic number theory techniques.

Contribution

It introduces a general, accessible criterion for the existence of infinitely many distinct zeros of two L-functions satisfying a functional equation, applicable beyond automorphic forms.

Findings

Establishes a criterion for zeros of two L-functions to be distinct

Applies to L-functions satisfying a functional equation and natural assumptions

Accessible approach not relying on advanced analytic number theory

Abstract

In this paper, we establish a simple criterion for two $L$-functions $L_1$ and $L_2$ satisfying a functional equation (and some natural assumptions) to have infinitely many distinct zeros. Some related questions have already been answered in the particular case of Automorphic forms using so-called Converse Theorems. Deeper results can also be stated for elements of the Selberg class. However, we shall give here a general answer that do not use any advanced topics in analytic number theory. Therefore, this paper should be accessible to anyone who has some basic notions in measure-theory and advanced complex analysis.