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.

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References in corpus (1)

• X-Ray of Riemann zeta-function

Cited by in corpus (1)

• Variants of the Riemann zeta function