An equation satisfied by all non-trivial zeros $\rho$ of the Riemann zeta function $\zeta$

TL;DR

The paper derives a new equation relating all non-trivial zeros of the Riemann zeta function, involving an integral and fractional parts, providing a novel analytical expression for these zeros.

Contribution

It presents a new equation satisfied by all non-trivial zeros of the Riemann zeta function, linking zeros to an integral expression involving fractional parts.

Findings

Derived an explicit equation for non-trivial zeros of ζ

Established a connection between zeros and integral expressions

Provides a new analytical tool for studying ζ zeros

Abstract

We show that if $\rho$ is a non-trivial zero of the Riemann zeta function $\zeta$ then $$2^\rho + \frac{1}{\rho - 1} + 1/2 = \rho \int_{1}^{\infty} {t + 1/2} t^{-\rho-1} dt$$ where, ${x}$ is the fractional part of $x$.