On an inequality for the Riemann zeta-function in the critical strip

TL;DR

This paper provides an elementary proof of a key inequality for the Riemann zeta-function within the critical strip using new power inequalities, and explores conditions related to the Riemann hypothesis.

Contribution

It introduces a novel elementary proof of the inequality |z(1-s)|  |z(s)| and proposes a new sufficient condition for the Riemann hypothesis involving derivatives of |z(s)|^2.

Findings

Proved the inequality |z(1-s)|  |z(s)| in the critical strip.

Established a sufficient condition for the Riemann hypothesis.

Conjectured the necessity of this condition.

Abstract

By using new power inequalities we give an elementary proof of an important relation for the Riemann zeta-function |\zeta(1-s)| <= |\zeta(s)| in the strip 0< Re s<1/2,\ |\Im s| >= 12. Moreover, we establish a sufficient condition of the validity of the Riemann hypothesis in terms of the derivative with respect to Re s of |\zeta(s)|^2 and conjecture its necessity.