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

2017/08/03 by Frank Stenger, Stenger, Frank · 1 voice
Mathematics · Physics and Astronomy · #11M06 #42A38 #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #General Mathematics (math.GM) #Mathematics and Applications #Meromorphic and Entire Functions #math.GM

paper · pdf · doi:10.48550/arxiv.1708.01209

openalex publication_date 2017/08/03 · arxiv published 2017/08/03 · openalex created_date 2022/10/14 · arxiv updated 2024/07/29 · openalex updated_date 2026/07/28

Abstract

In this paper we study the function G(z) := int0,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.

Discussions

• Proof of Riemann-Hypothesis [hn, 2 points, 1 comments]

Related

• On zeros of the Riemann zeta function
• An investigation of the non-trivial zeros of the Riemann zeta function
• All Complex Zeros of the Riemann Zeta Function Are on the Critical Line:\n Two Proofs of the Riemann Hypothesis
• Almost all of the nontrivial zeros of the Riemann zeta-function are on the critical line
• Notes on the Zeros of Riemann's Zeta Function