  In this paper we prove trace formulae for the Reidemeister number of a group
endomorphism. This result implies the rationality of the Reidemeister zeta
function in the following cases: the group is a direct product of a finite
group and a finitely generated Abelian group; the group is finitely generated,
nilpotent and torsion free. We continue to study analytical properties of the
Nielsen zeta function. We connect the Reidemeister zeta function of an
endomorphism of a direct product of a finite group and a finitely generated
free Abelian group with the Lefschetz zeta function of the induced map on the
unitary dual of the group. As a consequence we obtain a relation between a
special value of the Reidemeister zeta function and a certain Reidemeister
torsion. We also prove congruences for Reidemeister numbers of iterates of an
endomorphism of a direct product of a finite group and a finitely generated
free Abelian group which are the same as those found by Dold for Lefschetz
numbers.
