Proof of Riemann's zeta-hypothesis

TL;DR

This paper proposes a novel approach using integral transformations and residue calculus to derive series expansions of the Riemann zeta-function, aiming to prove that all nontrivial zeros lie on the critical line.

Contribution

It introduces a new method involving exponential transformations and residue calculus to analyze the zeta-function, providing a potential proof of the Riemann hypothesis.

Findings

Derived two equivalent series expansions valid in the critical strip

Showed the behavior of these expansions as N approaches infinity

Provided a framework suggesting all nontrivial zeros have real part 1/2

Abstract

Make an exponential transformation in the integral formulation of Riemann's zeta-function zeta(s) for Re(s) > 0. Separately, in addition make the substitution s -> 1 - s and then transform back to s again using the functional equation. Using residue calculus, we can in this way get two alternative, equivalent series expansions for zeta(s) of order N, both valid inside the "critical strip", i e for 0 < Re(s) < 1. Together, these two expansions embody important characteristics of the zeta-function in this range, and their detailed behavior as N tends to infinity can be used to prove Riemann's zeta-hypothesis that the nontrivial zeros of the zeta-function must all have real part 1/2. In addition to the preprint, the arXiv file also contains a discussion of some forty Frequently Asked Questions from readers. Further questions not adequately dealt with in the existing FAQ are welcome.