Fundamental Domains of Gamma and Zeta Functions

TL;DR

This paper investigates the fundamental domains of the Euler Gamma and Riemann Zeta functions on branched covering Riemann surfaces, revealing their symmetry groups and analyzing zero multiplicities through geometric and visualization techniques.

Contribution

It provides explicit fundamental domains and the group of covering transformations for both functions, and addresses the zero multiplicity problem for the Zeta function.

Findings

Fundamental domains for Gamma and Zeta functions are explicitly identified.

The group of covering transformations for both functions is characterized.

The zero multiplicity of the Riemann Zeta function and its derivative is determined.

Abstract

Branched covering Riemann surfaces $(\mathbb{C},f)$ are studied, where $f$ is the Euler Gamma function and the Riemann Zeta function. For both of them fundamental domains are found and the group of covering transformations is revealed. In order to find fundamental domains, pre-images of the real axis are taken and a thorough study of their geometry is performed. The technique of simultaneous continuation, introduced by the authors in previous papers, is used for this purpose. Color visualization of the conformal mapping of the complex plane by these functions is used for a better understanding of the theory. For the Riemann Zeta function the outstanding question of the multiplicity of its zeros, as well as of the zeros of its derivative is answered.