The $a$-points of the Riemann zeta-function and the functional equation

TL;DR

This paper establishes an equivalent formulation of the Riemann hypothesis using the functional equation and the roots of the zeta-function for any fixed complex number, linking the distribution of these roots to the hypothesis.

Contribution

It introduces a new criterion for the Riemann hypothesis based on the properties of the zeta-function's $a$-points and its functional equation.

Findings

Equivalent condition for the Riemann hypothesis involving $a$-points

Connection between the functional equation and root distribution

New perspective on the roots of the zeta-function

Abstract

We prove an equivalent of the Riemann hypothesis in terms of the functional equation (in its asymmetrical form) and the $a$-points of the zeta-function, i.e., the roots of the equation $\zeta(s)=a$, where $a$ is an arbitrary fixed complex number.