A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis

TL;DR

The paper proves a monotonicity property of Riemann's xi function in certain half-planes and offers a new reformulation of the Riemann Hypothesis based on this property.

Contribution

It establishes a novel monotonicity characterization of Riemann's xi function and reformulates the Riemann Hypothesis accordingly.

Findings

Riemann's xi function is strictly increasing in modulus along certain half-lines.

Riemann's xi function is strictly decreasing in modulus along other half-lines.

Provides a new equivalent condition for the Riemann Hypothesis.

Abstract

We prove that Riemann's xi function is strictly increasing (respectively, strictly decreasing) in modulus along every horizontal half-line in any zero-free, open right (respectively, left) half-plane. A corollary is a reformulation of the Riemann Hypothesis.