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

Source 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.