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arXiv · 1708.02671

Variants of the Riemann zeta function

Barry Brent
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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.

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Barry Brent. 2017-08-08. Variants of the Riemann zeta function. https://arxiv.org/abs/1708.02671

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