  Riemannian Geometry, Topology and Dynamics permit to introduce partially
defined holomorphic functions on the variety of representations of the
fundamental group of a manifold. The functions we consider are the complex
valued Ray-Singer torsion, the Milnor-Turaev torsion, and the dynamical
torsion. They are associated essentially to a closed smooth manifold equipped
with a (co)Euler structure and a Riemannian metric in the first case, a smooth
triangulation in the second case, and a smooth flow of type described in
section 2 in the third case. In this paper we define these functions, describe
some of their properties and calculate them in some case. We conjecture that
they are essentially equal and have analytic continuation to rational functions
on the variety of representations. We discuss the case of one dimensional
representations and other relevant situations when the conjecture is true. As
particular cases of our torsions, we recognize familiar rational functions in
topology such as the Lefschetz zeta function of a diffeomorphism, the dynamical
zeta function of closed trajectories, and the Alexander polynomial of a knot. A
numerical invariant derived from Ray-Singer torsion and associated to two
homotopic acyclic representations is discussed in the last section.
