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The domain of the Riemann Zeta function and that of its derivative appear as branched covering Riemann surfaces (C, f). The fundamental domains, which are the leafs of those surfaces are revealed. For this purpose, pre-images of the real axis by the two functions are taken and a thorough study of their geometry is performed. The study of intertwined curves generated in this way, allowed us to prove that the Riemann Zeta function has only simple zeros and finally that the Riemann Hypothesis is true. A version of this paper containing color visualization of the conformal mappings of the fundamental domains can be found in [10].

Mathematics Subject Classification (2010): 11M26, 30C55.

Keywords: Fundamental domain, branched covering Riemann surface, simultaneouscontinuation, Zeta function, non trivial zero.