Paper 2024/1748

New Experimental Evidences For the Riemann Hypothesis

Abstract

The zeta function $\zeta (z)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^z}$ is convergent for $\text{Re}(z)>1$, and the eta function $\eta (z)=\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n-1}}{n^z}$ is convergent for $\text{Re}(z)>0$. The eta function and the analytic continuation of zeta function have the same zeros in the critical strip $0<\text{Re}(z)<1$, owing to that $\eta (z)=(1-2^{1-z})\zeta (z)$. In this paper, we present the new experimental evidences which show that for any $a\in (0,1),b\in (-\mathrm{\infty },\mathrm{\infty })$, there exists a zero $\frac{1}{2}+it$ such that the modulus $|\eta (a+ib)|\ge |\eta (a+it)|>|\eta (\frac{1}{2}+it)|=0$. These evidences further confirm that all zeros are on the critical line $\text{Re}(z)=\frac{1}{2}$.

Note: This is a new version.