  We generalize to higher algebraic $K$-theory an identity (originally due to
Milnor) that relates the Reidemeister torsion of an infinite cyclic cover to
its Lefschetz zeta function. Our identity involves a higher torsion invariant,
the endomorphism torsion, of a parametrized family of endomorphisms as well as
a higher zeta function of such a family. We also exhibit several examples of
families of endomorphisms having non-trivial endomorphism torsion.
