  We introduce new zeta functions related to an endomorphism $\phi$ of a
discrete group $\Gamma$. They are of two types: counting numbers of fixed
($\rho\sim \rho\circ\phi^n$) irreducible representations for iterations of
$\phi$ from an appropriate dual space of $\Gamma$ and counting Reidemeister
numbers $R(\phi^n)$ of different compactifications. Many properties of these
functions and their coefficients are obtained. In many cases it is proved that
these zeta functions coincide. The Gauss congruences are proved. Useful
asymptotic formulas for the zeta functions are found. Rationality is proved for
some examples, which give also the first counterexamples simultaneously for
TBFT ($R(\phi)$=the number of fixed irreducible unitary representations) and
TBFT$_f$ ($R(\phi)$=the number of fixed irreducible unitary finite-dimensional
representations) for an automorphism $\phi$ with $R(\phi)<\infty$.
