Variants of the Riemann zeta function

TL;DR

This paper introduces variants of the Riemann zeta function, studies their fixed points and conjectures about their dynamics, potentially offering new insights into the relationship between zeta fixed points and Riemann zeros.

Contribution

It constructs and analyzes new variants of the zeta function with specific properties, proposing conjectures based on their dynamical behavior and geometric structures.

Findings

Fixed points of $V_z$ converge along logarithmic spirals

Spirals originate at zeta fixed points and center on Riemann zeros

Numerical approximations suggest a geometric link between fixed points and zeros

Abstract

We construct variants of the Riemann zeta function with convenient properties and make conjectures about their dynamics; some of the conjectures are based on an analogy with the dynamical system of zeta. More specifically, we study the family of functions $V_z: s \mapsto \zeta(s) \exp (zs)$. We observe convergence of $V_z$ fixed points along nearly logarithmic spirals with initial points at zeta fixed points and centered upon Riemann zeros. We can approximate these spirals numerically, so they might afford a means to study the geometry of the relationship of zeta fixed points to Riemann zeros.