Variants of the Riemann zeta function

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.

Repaired my omission of the definition of ${\Phi }_z$. 23 pages, 12 figures

References in corpus (2)

• X-Ray of Riemann zeta-function
• Experiments with the dynamics of the Riemann zeta function