Paper 2024/1412

A New 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 $\text{Re}(z)>1$. 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 $\frac{1}{2}+it_1$ and $\frac{1}{2}+it_2$. This method is non-polynomial time, because it has to compute the sum $\sum_{n\leq \alpha}\frac{\cos(\vartheta(1/2+it)-t\log{n})}{\sqrt{n}}$, where $\alpha=\lfloor\sqrt{t/(2\pi)}\rfloor$. The eta function $\eta(z)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^z}$ is convergent for $\text{Re}(z)>0$, and $\eta(z)=\left(1-2^{1-z}\right)\zeta(z)$ 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.