Asymmetry for the Riemann Hypothesis

TL;DR

This paper proves that the Riemann zeta function and its reflection do not both vanish off the critical line, supporting the Riemann Hypothesis, even under a generalized rotation hypothesis.

Contribution

It establishes a new non-vanishing property of the zeta function in the critical strip, extending previous results under a novel rotation number hypothesis.

Findings

The zeta function and its conjugate do not simultaneously vanish off the critical line.

This non-vanishing property holds even with a generalized fractional part function.

Supports the Riemann Hypothesis by restricting zeros to the critical line.

Abstract

In this manuscript, we show that the Riemann zeta function satisfies $\big(\zeta(s),\zeta(1-\overline{s})\big)\neq(0,0)$ for any $s$ in the critical strip, except on the critical line. This still holds even when the fractional part function is replaced by a function satisfying a specific "Rotation Number Hypothesis".