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

Source abstract

The Riemann Hypothesis (RH) is proved via a new expression of the completed Riemann zeta function $\xi(s)$, obtained through pairing the conjugate zeros $\rho_i$ and $\bar{\rho}_i$ in the Hadamard product while accounting for zero multiplicities (which are uniquely determined, although their specific values remain unknown), 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 $\xi(0)=\frac{1}{2}$, $\rho_i=\alpha_i+j\beta_i$, $\bar{\rho}_i=\alpha_i-j\beta_i$, 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 $\rho_i/\bar\rho_i$. Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have $$\prod_{i=1}^{\infty}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}}=\prod_{i=1}^{\infty}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}}$$ which, owing to the divisibility of entire functions, uniqueness of $m_i$, and the irreducibility of each real quadratic 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 RH is true.