  We study the Dirac and the Laplacian operators on orientable Riemann surfaces
of arbitrary genus g. In particular we compute their determinants with twisted
boundary conditions along the b-cycles. All the ingredients of the final
results (including the normalizations) are explicitly written in terms of the
Schottky parametrization of the Riemann surface. By using the bosonization
equivalence, we derive a multi-loop generalization of the well-known g=1
product formulae for the Theta-functions. We finally comment on the
applications of these results to the perturbative theory of open charged
strings.
