preprints.org > computer science and mathematics > algebra and number theory > doi: 10.20944/preprints202108.0146.v26

Preprint Article Version 26 Preserved in Portico This version is not peer-reviewed

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The celebrated Riemann Hypothesis (RH) is proved based on a new absolute convergent expression of $\xi(s)$, which was obtained from the Hadamard product, through paring $\rho_i$ and $\bar{\rho}_i$, and putting all the multiple zeros together in one factor, i.e. $$\xi(s)=\xi(0)\prod_{\rho}(1-\frac{s}{\rho})=\xi(0)\prod_{i=1}^{\infty}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}^{d_{i}}$$ where $\xi(0)=\frac{1}{2}$, $\rho_i=\alpha_i+j\beta_i$ 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, $d_i\geq 1$ is the real (\textbf{unique and unchangeable}) multiplicity of $\rho_i$, $\beta_i$ are arranged in order of increasing $|\beta_i|$, i.e., $|\beta_1|\leq|\beta_2|\leq|\beta_3|\leq \cdots$, $i =1,2,3, \cdots, \infty$.\\ Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have $$\prod_{i=1}^{\infty}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)}^{d_{i}}=\prod_{i=1}^{\infty}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}^{d_{i}}$$ which, owing to the uniqueness and unchangeableness of $d_i$ (see Lemma 3 for the proof details), is finally equivalent to $$\begin{cases}&\alpha_i=\frac{1}{2}\\ & |\beta_1|<|\beta_2|<|\beta_3|<\cdots\\&i =1,2,3, \cdots, \infty \end{cases}$$ Thus, we conclude that the RH is true.

Riemann Hypothesis (RH); proof; completed zeta function

Computer Science and Mathematics, Algebra and Number Theory

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