Proof of Riemann hypothesis by topological and analytical methods

Abstract

We introduce a differential topological proof and an analytical proof of Riemann hypothesis according to the saddle point method because Riemann calculated the integral representation of zeta function on the critical line by this method. This topological proof of RH proves that the existence of integral representation of zeta functions requires certain differential topological conditions on its integrand according to which the zeta function vanishes on the critical line. The analytical proof of RH is the local implementation of topological proof or its coordinate representation.