Experiments with the dynamics of the Riemann zeta function

TL;DR

This paper presents experimental evidence on the geometric and dynamical properties of the Riemann zeta function, revealing spiral structures and orbit behaviors related to its zeros and fixed points.

Contribution

It introduces new empirical observations about the spiral dynamics and orbit structures associated with the Riemann zeta function and its zeros.

Findings

Existence of nearly logarithmic spirals centered at fixed points containing zeros.

Identification of subsets of backward zeta orbits with specific properties.

Nearly uniform angular distribution of certain orbit sets.

Abstract

We collect experimental evidence for several propositions, including the following: (1) For each Riemann zero $\rho$ (trivial or nontrivial) and each zeta fixed point $\psi$ there is a nearly logarithmic spiral $s_{\rho, \psi}$ with center $\psi$ containing $\rho$. (2) $s_{\rho, \psi}$ interpolates a subset $B_{\rho, \psi}$ of the backward zeta orbit of $\rho$ comprising a set of zeros of all iterates of zeta. (3) If zeta is viewed as a function on sets, $\zeta(B_{\rho, \psi}) = B_{\rho, \psi} \cup \{0 \}$. (4) $B_{\rho, \psi}$ has nearly uniform angular distribution around the center of $s_{\rho, \psi}$. We will make these statements precise.