Experiments with the dynamics of the Riemann zeta function

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.