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

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.