Notes on the Zeros of Riemann's Zeta Function

TL;DR

This paper investigates the zeros of Riemann's Zeta function, providing a proof that all non-trivial zeros lie on the critical line and deriving equations to locate these zeros.

Contribution

It offers a proof confirming the Riemann Hypothesis and introduces equations for precisely locating all zeros on the critical line.

Findings

All non-trivial zeros of ζ(s) occur on the critical line.

Derived transcendental equations can locate zeros accurately.

Confirmed the truth of Riemann's Hypothesis.

Abstract

The functional equation for Riemann's Zeta function is studied, from which it is shown why all of the non-trivial, full-zeros of the Zeta function $\zeta (s)$ will only occur on the critical line {$\sigma=1/2$} where {$s=\sigma+I \rho$}, thereby establishing the truth of Riemann's hypothesis. Further, two relatively simple transcendental equations are obtained; the numerical solution of these equations locates all of the zeros of {$\zeta (s)$} on the critical line.