  For an acyclic representation of the fundamental group of a compact oriented
odd-dimensional manifold, which is close enough to a unitary representation, we
define a refinement of the Ray-Singer torsion associated to this
representation. This new invariant can be viewed as an analytic counterpart of
the refined combinatorial torsion introduced by Turaev.
  The refined analytic torsion is a holomorphic function of the representation
of the fundamental group. When the representation is unitary, the absolute
value of the refined analytic torsion is equal to the Ray-Singer torsion, while
its phase is determined by the eta-invariant. The fact that the Ray-Singer
torsion and the eta-invariant can be combined into one holomorphic function
allows to use methods of complex analysis to study both invariants. In
particular, we extend and improve a result of Farber about the relationship
between the Farber-Turaev absolute torsion and the eta-invariant.
