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A Proof of the Riemann Hypothesis Based on a New Expression of the Completed Zeta Function
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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.v28 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.v28
Abstract
The celebrated Riemann Hypothesis (RH) is proved based on a new absolute convergent expression of $\xi(s)$, which was obtained from the Hadamard product, through paring $\rho_i$ and $\bar{\rho}_i$, and putting all the multiple zeros together in one factor, i.e. $$\xi(s)=\xi(0)\prod_{\rho}(1-\frac{s}{\rho})=\xi(0)\prod_{i=1}^{\infty}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}^{d_{i}}$$ 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, $d_i\geq 1$ is the real (unique and unchangeable) multiplicity of $\rho_i$, $\beta_i$ are arranged in order of increasing $|\beta_i|$, i.e., $|\beta_1|\leq|\beta_2|\leq|\beta_3|\leq \cdots$, $i =1,2,3, \cdots, \infty$. Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have $$\prod_{i=1}^{\infty}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)}^{d_{i}}=\prod_{i=1}^{\infty}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}^{d_{i}}$$ which, owing to the uniqueness and unchangeableness of $d_i$ (see Lemma 3 for the proof details), is finally equivalent to $$\begin{cases}&\alpha_i=\frac{1}{2}\\ & |\beta_1|<|\beta_2|<|\beta_3|<\cdots\\&i =1,2,3, \cdots, \infty \end{cases}$$ Thus, we conclude that the RH is true.
Keywords
Riemann Hypothesis (RH); proof; completed zeta function
Subject
Computer Science and Mathematics, Algebra and 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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Commenter: Weicun Zhang
Commenter's Conflict of Interests: Author
1. The proof of Lemm 3 has been updated, especially the Limit Case;
2. The proof of Lemma 4 has been updated, especially the explaination on the solution of equation (28).