A Proof of the Riemann Hypothesis Based on a New Expression of the Completed Zeta Function
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The Riemann Hypothesis (RH) is proved based on a new expression of the completed zeta function \( \xi(s) \), which was obtained by pairing the conjugate zeros in the Hadamard product, and considering the multiplicity of zeros. Specifically, we have: \( \xi(s)=\xi(0)\prod_{\rho}(1-\frac{s}{\rho})=\xi(0)\prod_{i=1}^{\infty}(1-\frac{s}{\rho_i})(1-\frac{s}{\bar{\rho}_i})=\xi(0)\prod_{i=1}^{\infty}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}^{m_{i}} \) where \( \xi(0)=\frac{1}{2}, \rho_i=\alpha_i+j\beta_i, \bar{\rho}_i=\alpha_i-j\beta_i \), with \( 0<\alpha_i<1, \beta_i\neq 0, 0<|\beta_1|\leq|\beta_2|\leq \cdots \), and \( m_i\geq 1 \) is the multiplicity of \( \rho_i \). Then, according to the functional equation \( \xi(s)=\xi(1-s) \), we have \( \prod_{i=1}^{\infty}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}}=\prod_{i=1}^{\infty}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}} \) Due to the divisibility contained in the above equation and the uniqueness of \( m_i \), each polynomial factor can only divide (and thus equal) the corresponding factor on the opposite side of the equation. Therefore, we finally obtain \( \alpha_i=\frac{1}{2}, 0<|\beta_1|<|\beta_2|<|\beta_3|<\cdots, i=1, 2, 3, \dots, \infty \) Thus, we conclude that the RH is true.