A Proof of the Riemann Hypothesis Based on a New Expression of ξ(s)

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.