Paper 2024/1748

A Simple Method to Test the Zeros of Riemann Zeta Function

Abstract

The zeta function $\zeta(z)=\sum_{n=1}^{\infty} \frac{1}{n^z}$ is convergent only for $\text{Re}(z)>1$. The Riemann-Siegel function is $Z(t)=e^{i\vartheta(t)}\zeta(\frac{1}{2}+it)$. If $Z(t_1)$ and $Z(t_2)$ have opposite signs, $Z(t)$ vanishes between $t_1$ and $t_2$, and $\zeta(z)$ has a zero on the critical line between $\frac{1}{2}+it_1$ and $\frac{1}{2}+it_2$. This method to test zeros is too hard to practice for newcomers. The eta function $\eta(z)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^z}$ is convergent for $\text{Re}(z)>0$, and $\eta(z)=\left(1-\frac{2}{2^z}\right)\zeta(z)$ for the critical strip $0<\text{Re}(z)<1$. So, $\eta(z)$ and the analytic continuation of $\zeta(z)$ have the same zeros in the critical strip, and the alternating series can be directly used to test the zeros.