  We prove a dichotomy between rationality and a natural boundary for the
analytic behavior of the Reidemeister zeta function for automorphisms of
non-finitely generated torsion abelian groups and for endomorphisms of groups
$\mathbb Z_p^d,$ where $\mathbb Z_p$ the group of p-adic integers. As a
consequence, we obtain a dichotomy for the Reidemeister zeta function of a
continuous map of a topological space with fundamental group that is
non-finitely generated torsion abelian group. We also prove the rationality of
the coincidence Reidemeister zeta function for tame endomorphisms pairs of
finitely generated torsion-free nilpotent groups, based on a weak commutativity
condition.
