An experimental study of the monotonicity property of the Riemann zeta function

TL;DR

This paper experimentally investigates the conjectured monotonicity of the Riemann zeta function's magnitude, revealing new properties and behaviors of z(z) and related functions.

Contribution

It provides the first experimental analysis of Spira's monotonicity conjecture and introduces new properties of z(z), including the spectrum of semi-limits and the core function.

Findings

Evidence supporting the monotonicity conjecture in certain regions

Discovery of the spectrum of semi-limits z(z)

Introduction of the core function C(z) as a simplified model

Abstract

In 1970, based on newly available empiric evidence, a remarkable monotonicity property for $| \zeta(z) |$ was conjectured by R. Spira. The $\zeta$-monotonicity property can be written as follows: $$ | \zeta (x_2 + y i ) | < | \zeta \left ( x_1 +y i \right )| \hspace{0.5cm} \textrm {for any } \hspace{0.25cm} x_1 < x_2 \leq 0.5 \textrm{ and } 6.29 <y. $$ In this work we present an experimental study of the monotonicity conjecture, in the course of which new properties of $\zeta(z)$ are discovered. For instance, the spectrum of semi-limits $ \lambda(z) \subset \mathbb{R}$ and the core function $C(z)$, which serves as a non-chaotic simplification of $\zeta(z)$ to the left of the critical line