The Reidemeister zeta functions and the Gauss congruences for the Reidemeister numbers
Volume 181 / 2026
Colloquium Mathematicum 181 (2026), 41-61
MSC: Primary 37C25; Secondary 37C30, 20E45, 55M20
DOI: 10.4064/cm9590-12-2025
Published online: 14 July 2026
Abstract
We prove a dichotomy between rationality and a natural boundary for the analytic behavior of the Reidemeister zeta function for tame endomorphisms of $\mathbb Z _p^d,$ where $\mathbb Z _p$ is the additive group of $p$-adic integers. We also prove the rationality of the coincidence Reidemeister zeta function for tame endomorphism pairs of finitely generated torsion-free nilpotent groups, based on a weak commutativity condition. Furthermore, we prove the Gauss congruences for Reidemeister coincidence numbers of iterations of tame endomorphism pairs of finitely generated torsion-free nilpotent groups.
