All Zeros of the Riemann Zeta Function in the Critical Strip are Located on the Critical Line and are Simple

TL;DR

This paper introduces a new function G(z) related to the Riemann Zeta function, proves they share zeros in the critical strip, and claims all zeros on the critical line are simple, supporting the Riemann hypothesis.

Contribution

It establishes a functional relation between G(z) and the Zeta function, showing they share zeros and that zeros on the critical line are simple, providing new insights into the Riemann hypothesis.

Findings

G(z) and Zeta share zeros in the critical strip

All zeros on the critical line are simple

Supports the Riemann hypothesis that zeros lie on the critical line

Abstract

In this paper we study the function G(z) := int{0,infinity} y^{z-1}(1 + \exp(y))^{-1} dy, for z in C. We derive a functional equation that relates G(z) and G(1 - z) for all z in C, and we prove: -- That G and the Riemann Zeta function Zeta have exactly the same zeros in the critical region D := z in C: Re z in (0,1); -- All the zeros of the Riemann Zeta function located on the critical line are simple; and -- The Riemann hypothesis, i.e., that all of the zeros of G in D are located on the critical line L := {z in D : Re z = 1/2}.