On the Non-Trivial Zeros of Dirichlet Functions

Dorin Ghisa

doi:10.4236/apm.2021.113014

Abstract

The purpose of this research is to extend to the functions obtained by meromorphic continuation of general Dirichlet series some properties of the functions in the Selberg class, which are all generated by ordinary Dirichlet series. We wanted to put to work the powerful tool of the geometry of conformal mappings of these functions, which we built in previous research, in order to study the location of their non-trivial zeros. A new approach of the concept of multiplier in Riemann type of functional equation was necessary and we have shown that with this approach the non-trivial zeros of the Dirichlet function satisfying a Reimann type of functional equation are either on the critical line, or they are two by two symmetric with respect to the critical line. The Euler product general Dirichlet series are defined, a wide class of such series is presented, and finally by using geometric and analytic arguments it is proved that for Euler product functions the symmetric zeros with respect to the critical line must coincide.

Highlights

The fundamental domains, as defined by Ahlfors, are for the complex functions of one complex variable the similar of the intervals of monotony for the real ones, in the sense that in both cases the functions are injective there
We have proved in [16]: Proposition 3: If ζ A,Λ (s) satisfies a Riemann type of functional equation, but does not satisfy the Riemann Hypothesis (RH), : (a) The off critical line non-trivial zeros appear in couples of the form s=1 σ1 + it and s2 =σ 2 + it =1− σ1 + it, where σ1 > 1 2
In order to properly tackle the extension of the Riemann Hypothesis to a class of functions generated by general Dirichlet series, we needed to reformulate the concept of trivial and non-trivial zeros

Summary

Introduction

The fundamental domains, as defined by Ahlfors, are for the complex functions of one complex variable the similar of the intervals of monotony for the real ones, in the sense that in both cases the functions are injective there. For the analytic functions of one complex variable the concept of fundamental domain plays a much deeper role. Such a domain is mapped conformally by the function onto the whole complex plane with some slits. The rational functions have a finite number of fundamental domains (the same as their degree), while the transcendental ones have infinitely many fundamental domains. Their closure covers the entire complex plane

Ghisa DOI

Meromorphic Continuation of General Dirichlet Series

The Geometry of Mappings by Dirichlet Functions

The Zeros of Dirichlet Functions

Conclusion