Fundamental Domains of Gamma and Zeta Functions

Abstract

Branched covering Riemann surfaces(ℂ,f)are studied, wherefis the Euler Gamma function and the Riemann Zeta function. For both of them fundamental domains are found and the group of cover transformations is revealed. In order to find fundamental domains, preimages 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. A version of this paper containing colored images can be found in arXiv at Andrian Cazacu and Ghisa.

On The Generalized Riemann Hypothesis II

“…The concept of fundamental domain coming from the automorphic functions theory and to which Ahlfors [5] gave an independent meaning helped us to fulfill this task and to advance the research in the field of Dirichlet series. The Utzinger and Speiser reellen Züge were used only to put into evidence in [6], [7], [8], [9] the boundaries of fundamental domains of the class of functions we were studying. This class of functions includes the Selberg class, for which the most general RH has been formulated.…”