Majoration du Nombre de Zéros D’Une Fonction Méromorphe 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) ± h(2a − s), where h(s) is a meromorphic function, real on the real line, a is 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 ℜs = a, called the critical line for the function f(s), and that they are simple, provided that all but finitely many of the zeros of h(s) lie on the half-plane ℜs < a. This result can be regarded as a generalization of the necessary condition of stability for the function h(s), in the Hermite-Biehler theorem. We apply our results to the study of translations of the Riemann Zeta Function and L functions, and integrals of Eisenstein Series, among others.