On Studying the Phase Behavior of the Riemann Zeta Function Along the Critical Line

TL;DR

This paper investigates the phase behavior of the Riemann zeta function along the critical line, revealing a unique phase relation between zeros and conjugate points, and deriving an equation for the phase based on the functional equation.

Contribution

It introduces a novel perspective on the phase properties of the zeta function along the critical line and derives a new equation for its phase using the functional equation.

Findings

The ratio of the zeta function at zeros and conjugate points has a specific phase with magnitude one.

This phase relation holds exclusively along the critical line.

An explicit equation for the phase along the critical line is derived.

Abstract

The critical line of the Riemann zeta function is studied from a new viewpoint. It is found that the ratio between the zeta function at any zero and the corresponding one at a conjugate point has a certain phase and its absolute value is unity. This fact is valid along the whole critical line and only there. The common functional equation is used with the aid of the function ratio between any zero and its negative side pair, a complex conjugate. As a result, an equation is obtained for solving the phase along the critical line.