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 through paring the conjugate zeros $\rho_i$ and $\bar{\rho}_i$ in the Hadamard product, with consideration of the multiplicity of zeros, i.e. $$\xi(s)=\xi(0)\prod_{\rho}(1-\frac{s}{\rho})=\xi(0)\prod_{i=1}^{\infty}(1-\frac{s}{\rho_i})(1-\frac{s}{\bar{\rho}_i})=\xi(0)\prod_{i=1}^{\infty}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}^{m_{i}}$$ 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, $m_i\geq 1$ is the multiplicity of $\rho_i$, finite and unique, $0<|\beta_1|\leq|\beta_2|\leq \cdots$. Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have $$\prod_{i=1}^{\infty}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}}=\prod_{i=1}^{\infty}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}}$$ Owing to the divisibility contained in the above equation and the uniqueness of $m_i$, it is equivalent to $$\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}}=\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}}, i=1, 2, 3, \dots, \infty $$ which is further equivalent to $$\alpha_i=\frac{1}{2}, 0<|\beta_1|<|\beta_2|<|\beta_3|<\cdots, i=1, 2, 3, \dots, \infty$$ Thus we conclude that the RH is true.