The Riemann Hypothesis Is True If The Zeros Of \(\xi(s)\) Have Strict Increasing Imaginary Parts

Based on Hadamard product, a new expression of the completed zeta function \(\xi(s)\) is obtained, i.e., \[\xi(s)=\xi(0)\prod_{i=1}^{\infty}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}^{d_i}\] where \(\rho_i=\alpha_i+j\beta_i, \bar{\rho}_i=\alpha_i-j\beta_i\) are complex conjugate zeros of \(\xi(s)\), \(d_i\geq 1\) (natural numbers) are the multiplicities of \(\rho_i\), \(0<\alpha_i<1, \beta_i\neq 0, i\in \mathbb{N}\) are natural numbers from 1 to infinity. Suppose \(\beta_i\) are in order of strict increasing $|\beta_i|$, i.e., $|\beta_1|<|\beta_2|<|\beta_3|<, \cdots$, or in another word, for each \(\beta_i\), there are only four (two pairs) possible zeros (regardless of their multiplicities), i.e., \(\alpha_i \pm j\beta_i, 1-\alpha_i \pm j\beta_i\). According to the functional equation \(\xi(s)=\xi(1-s)\), We have \[\xi(0)\prod_{i=1}^{\infty}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}^{d_i} =\xi(0)\prod_{i=1}^{\infty}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(1-s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}^{d_i}\] i.e., \[\prod_{i=1}^{\infty}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)}^{d_i}=\prod_{i=1}^{\infty}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}^{d_i}\] which, by Lemma 3, is equivalent to \[\alpha_i= \frac{1}{2}, i\in \mathbb{N}, \text{ from 1 to infinity.}\] Thus, the Riemann Hypothesis is true if we have $|\beta_1|<|\beta_2|<|\beta_3|<, \cdots$.