A Proof of the Riemann Hypothesis Based on a New Expression of the Completed Zeta Function

Weicun Zhang

doi:10.20944/preprints202108.0146.v14

Preprint Article Version 14 Preserved in Portico This version is not peer-reviewed

A Proof of the Riemann Hypothesis Based on a New Expression of the Completed Zeta Function

Version 1 : Received: 2 August 2021 / Approved: 5 August 2021 / Online: 5 August 2021 (12:21:56 CEST)
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How to cite: Zhang, W. A Proof of the Riemann Hypothesis Based on a New Expression of the Completed Zeta Function. Preprints 2021, 2021080146 (doi: 10.20944/preprints202108.0146.v14). Zhang, W. A Proof of the Riemann Hypothesis Based on a New Expression of the Completed Zeta Function. Preprints 2021, 2021080146 (doi: 10.20944/preprints202108.0146.v14).

Abstract

The completed zeta function $\xi(s)$ is expanded in MacLaurin series (infinite polynomial), which can be further expressed as infinite product (Hadamard product) of quadratic factors by its complex conjugate zeros $\alpha_i\pm j\beta_i, \beta_i\neq 0, i\in \mathbb{N}$ are natural numbers, from $1$ to infinity, $\mathbb{N}$ is the set of natural numbers. Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have $$\xi(0)\prod_{i=1}^{\infty}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)} =\xi(0)\prod_{i=1}^{\infty}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}$$ which, by Lemma 3 and Corollary 1, is equivalent to $$(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}$$ with solution $\alpha_i= \frac{1}{2}, i\in \mathbb{N}$ (another solution $s=\frac{1}{2}$ is invalid due to obvious contradiction). Thus, a proof of the Riemann Hypothesis is achieved.

Keywords

Riemann Hypothesis (RH); Proof ; Completed zeta function

Subject

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Comments (1)

Comment 1

Received: 21 December 2021
Commenter: Zhang Weicun
Commenter's Conflict of Interests: Author

Comment: With this updated version, discussions about the convergence of the new infinite product expression of the completed zeta function, have been added.

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