  Let f be a Morse map from a closed manifold to a circle. S.P.Novikov
constructed an analog of the Morse complex for f. The Novikov complex is a
chain complex defined over the ring of Laurent power series with integral
coefficients and finite negative part. This complex depends on the choice of a
gradient-like vector field. The homotopy type of the Novikov complex is the
same as the homotopy type of the completed complex of the simplicial chains of
the cyclic covering associated to f. In the present paper we prove that for
every C^0-generic f-gradient there is a homotopy equivalence between these two
chain complexes, such that its torsion equals to the Lefschetz zeta-function of
the gradient flow. For these gradients the Novikov complex is defined over the
ring of rational functions and the Lefschetz zeta-function is also rational.
The paper contains also a survey of Morse-Novikov theory and of the previous
results of the author on the C^0-generic properties of the Novikov complex.
