Welcome to the resource topic for 2024/1748
Title:
A Simple Method to Test the Zeros of Riemann Zeta Function
Authors: Zhengjun Cao
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.
ePrint: https://eprint.iacr.org/2024/1748
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