Copyright © 2011 Cabiria Andreian Cazacu and Dorin Ghisa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Branched covering Riemann surfaces are studied, where is 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.