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

Weicun Zhang

تاريخ النشر: 30 أكتوبر 2023
الخادم: Preprints.org
DOI: 10.20944/preprints202108.0146.v27

الملخص

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.

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