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: $$|\zeta (x_2+yi)|<|\zeta \left(x_1+yi\right)|\text{for any }{x}_1<x_2\le 0.5\text{ and }6.29<y.$$ 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

References in corpus (1)

• Riemann zeta function and quantum chaos