Majoration du nombre de zéros d'une fonction méromorphe en dehors d'une droite verticale et applications

Details Majoration du nombre de zéros d'une fonction méromorphe en dehors d'une droite verticale et applications Majoration du nombre de zéros d'une fonction méromorphe en dehors d'une droite verticale et applications

2007-12-08

arxiv.org/abs/0712.1266v1

Mathematics

Number Theory

46 pages; 2 figures

Scientific paper

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.

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