Paper 2024/1412

A New Method to Test the Zeros of Riemann Zeta Function

Abstract

The zeta function $\zeta (z)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^z}$ is convergent only for $\text{Re}(z)>1$. To test its zeros, one needs to use the Riemann-Siegel function $Z(t)$. 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 $\frac{1}{2}+i{t}_1$ and $\frac{1}{2}+i{t}_2$. This method is non-polynomial time, because it has to compute the sum $\sum _{n\le \alpha }\frac{\cos (\vartheta (1/2+it)-t\log n)}{\sqrt{n}}$, where $\alpha =\lfloor \sqrt{t/(2\pi )}\rfloor$. The eta function $\eta (z)=\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n-1}}{n^z}$ is convergent for $\text{Re}(z)>0$, and $\eta (z)=(1-2^{1-z})\zeta (z)$ for the critical strip $0<\text{Re}(z)<1$. The alternating series can be directly used to test the zeros because $\eta (z)$ and the analytic continuation of $\zeta (z)$ have the same zeros in the critical strip. In this paper, we present a polynomial time algorithm to test the zeros based on $\eta (z)$, which is more understandable and suitable for modern computing machines than the general method. Besides, we clarify the actual meaning of logarithm symbol in the Riemann-Siegel formula.

Note: This is a new version.