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

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