On the Motion of Zeros of Zeta Functions

TL;DR

This paper investigates the movement of zeros of various zeta functions, proposing a distribution formula for Hurwitz zeta zeros, examining symmetric zero distributions in combined functions, and discussing the hypothetical non-trivial Riemann zeros off the critical line.

Contribution

It introduces an accurate zero distribution formula for Hurwitz zeta functions and explores zero symmetries in linear combinations, also discussing non-trivial zeros outside the critical line.

Findings

Distribution formula for Hurwitz zeta zeros

Symmetric zero distributions in combined functions

Discussion on non-trivial zeros off the critical line

Abstract

The motion in the complex plane of the zeros to various zeta functions is investigated numerically. First the Hurwitz zeta function is considered and an accurate formula for the distribution of its zeros is suggested. Then functions which are linear combinations of different Hurwitz zeta functions, and have a symmetric distribution of their zeros with respect to the critical line, are examined. Finally the existence of the hypothetical non-trivial Riemann zeros with $Re(s)\neq 1/2$ is discussed.