A Proof of the Riemann Hypothesis Based on a New Expression of ξ(s)
Discuss this preprint
Start a discussion What are Sciety discussions?Listed in
This article is not in any list yet, why not save it to one of your lists.Abstract
The Riemann Hypothesis (RH) is proved based on a new expression of the completed zeta function ξ(s), which was obtained through pairing the conjugate zero ρρi and ρi‾ in the Hadamard product, with consideration of zero multiplicity, i.e. \( \xi(s)=\xi(0)\prod_{\rho}(1-\frac{s}{\rho})=\xi(0)\prod_{i=1}^{\infty}(1-\frac{s}{\rho_i})(1-\frac{s}{\bar{\rho}_i})=\xi(0)\prod_{i=1}^{\infty}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}^{m_{i}} \), wheree \( \xi(0)=\frac{1}{2} \), \( \rho_i=\alpha_i+j\beta_i \), \( \bar{\rho}_i=\alpha_i-j\beta_i \), with \( 0<\alpha_i<1, \beta_i\neq 0, 0<|\beta_1|\leq|\beta_2|\leq \cdots \), and \( m_i ≥ 1 \) is the multiplicity of \( \rho_i \). Then, according to the functional equation \( \xi(s)=\xi(1-s) \), we obtain \( \prod_{i=1}^{\infty}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}}=\prod_{i=1}^{\infty}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}} \), which, owing to the divisibility of entire function, uniqueness of $m_i$, and the irreducibility of each polynomial factor, is finally equivalent to \( \alpha_i=\frac{1}{2}, 0<|\beta_1|<|\beta_2|<|\beta_3|<\cdots, i=1, 2, 3, \dots \) Thus, we conclude that the Riemann Hypothesis is true.