Proof of Riemann hypothesis by topological and analytical methods

TL;DR

This paper presents new topological and analytical proofs of the Riemann hypothesis, linking integral representations and differential topology to the zeros of the zeta function on the critical line.

Contribution

It introduces the first differential topological and analytical proofs of RH, connecting integral representations with topological conditions on the zeta function.

Findings

Proof that integral representations impose topological conditions on the zeta function.

Demonstration that these conditions imply zeros on the critical line.

Analytical proof aligns with the topological proof through local implementation.

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.