  In this paper we prove the trace formulas for the Reidemeister numbers of
group endomorphisms in the following cases:the group is finitely generated and
an endomorphism is eventually commutative; the group is finite ; the group is a
direct sum of a finite group and a finitely generated free abelian group; the
group is finitely generated, nilpotent and torsion free . These results had
previously been known only for the finitely generated free abelian group.As a
consequence, we obtain under the same conditions on the fundamental group of a
compact polyhedron,the trace formulas for the Reidemeister numbers of a
continuous map and under suitable conditions the trace formulas for the Nielsen
numbers of a continuous map.The trace formula for the Reidemeister numbers
implies the rationality of the Reidemeister zeta function. We prove the
rationality and functional equation for the Reidemeister zeta function of an
endomorphisms of finitely generated torsion free nilpotent group and of a
direct sum of a finite group and a finitely generated free abelian group.We
give a new proof of the rationality of the Reidemeister zeta function in the
case when the group is finitely generated and the endomorphism is eventually
commutative and in the case when group is finite.We give also another proof for
the positivity of the radius of convergence of the Nielsen zeta function and
propose an exast algebraic lower bound estimation for the radius.We connect the
Reidemeister zeta function of an endomorphism of a direct sum of a finite group
and a finitely generated free abelian group with the Lefschetz zeta function of
the unitary dual map,and as consequence obtain a connection of the Reidemeister
zeta function with Reidemeister torsion.
