A Proof of the Riemann Hypothesis Based on a New Expression of the Completed Zeta Function

Source abstract

The Riemann Hypothesis (RH) is proved based on a new absolutely convergent expression of $\xi(s)$, which was obtained from the Hadamard product, through paring $\rho_i$ and $\bar{\rho}_i$, and taking the possible multiple zeros into consideration with their real (unique and unchangeable) multiplicities, 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 real multiplicity of $\rho_i$, $0<|\beta_1|\leq|\beta_2|\leq|\beta_3|\leq \cdots$. Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have $$\prod_{i=1}^{\infty}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}}=\prod_{i=1}^{\infty}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}}$$ which, owing to the uniqueness and unchangeableness of $m_i$, is finally equivalent to (for more details, see the proof of Lemma 3.) $$\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}}=\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}} \Leftrightarrow \alpha_i=\frac{1}{2}, 0<|\beta_1|<|\beta_2|<|\beta_3|<\cdots$$ Thus, we conclude that the RH is true.