Preprint Article Version 22 Preserved in Portico This version is not peer-reviewed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. https://doi.org/10.20944/preprints202108.0146.v22 Zhang, W. A Proof Of The Riemann Hypothesis Based On A New Expression Of The Completed Zeta Function. Preprints 2021, 2021080146. https://doi.org/10.20944/preprints202108.0146.v22

Abstract

Based on the Hadamard product $\xi(s)= \xi(0)\prod_{\rho}(1-\frac{s}{\rho})$, a new expression of $\xi(s)$ is obtained by paring $\rho$ and $\bar{\rho}$ $$\xi(s)= \xi(0)\prod_{i=1}^{\infty}(1-\frac{s}{\rho_i})(1-\frac{s}{\bar{\rho}_i})=\xi(0)\prod_{i=1}^{\infty}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}$$ where $\xi(0)=\frac{1}{2}$, $\rho_i=\alpha_i+j\beta_i$ and $\bar{\rho}_i=\alpha_i-j\beta_i$ are the complex conjugate zeros of $\xi(s)$, $0<\alpha_i<1$ and $\beta_i\neq 0$ are real numbers, $\beta_i$ are in order of increasing $|\beta_i|$, i.e., $|\beta_1|\leq|\beta_2|\leq|\beta_3|\leq \cdots$.\\ Then, by the functional equation $\xi(s)=\xi(1-s)$, we have $$\xi(0)\prod_{i=1}^{\infty}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)} =\xi(0)\prod_{i=1}^{\infty}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(1-s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}$$ i.e., $$\prod_{i=1}^{\infty}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)}=\prod_{i=1}^{\infty}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}$$ which, by Lemma 3, is equivalent to $$\alpha_i= \frac{1}{2}, i =1, 2, 3, \cdots, \infty$$ Thus, we conclude that the Riemann Hypothesis is true.

Keywords

Riemann Hypothesis (RH); Proof ; Completed zeta function

Subject

Computer Science and Mathematics, Algebra and Number Theory

Comments (1)

Comment 1
Received: 12 October 2022
Commenter: Zhang Weicun
Commenter's Conflict of Interests: Author
Comment: With this updated version, Lemma 3 has been further simplified, and has become a new section (Section 2: A Key Lemma).
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