Majoration du nombre de z\'eros d'une fonction m\'eromorphe en dehors d'une droite verticale et applications

TL;DR

This paper investigates the zeros of certain meromorphic functions, establishing conditions for their zeros to predominantly lie on a critical line, with applications to the Riemann Zeta function, L-functions, and Eisenstein series.

Contribution

It generalizes the Hermite-Biehler theorem by providing conditions for zeros to lie on a critical line for functions of the form h(s) ± h(2a-s).

Findings

Most zeros of f(s) are on the line Re s = a under certain conditions.

Zeros of f(s) are simple when h(s) zeros are mostly on Re s < a.

Applications include zeros of translated Riemann Zeta and L-functions.

Abstract

We study the distribution of the zeros of functions of the form $f(s)=h(s) \pm h(2a-s)$, where $h(s)$ is a meromorphic function, real on the real line, $a$ a real number. One of our results establishes sufficient conditions under which all but finitely many of the zeros of $f(s)$ lie on the line $\Re s = a$, called the {\it critical line} for the function $f(s)$, and be simple, given that all but finitely many of the zeros of $h(s)$ lie on the half-plane $\Re s < a$. This results can be regarded as a generalization of the necessary condition of stability for the function $h(s)$, in the Hermite-Biehler theorem. We apply this results to the study of translations of the Riemann Zeta Function and $L$ functions, and integrals of Eisenstein Series, among others.