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

Frank Stenger

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arXiv · 1708.01209

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

Frank Stenger

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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}.

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BibTeX RIS

Frank Stenger. 2017-08-03. All Zeros of the Riemann Zeta Function in the Critical Strip are Located on the Critical Line and are Simple. https://arxiv.org/abs/1708.01209

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