  We present the computation of threshold functions for Abelian orbifold
compactifications. Specifically, starting from the massive, moduli-dependent
string spectrum after compactification, we derive the threshold functions as
target space duality invariant free energies (sum over massive string states).
In particular we work out the dependence on the continuous Wilson line moduli
fields. In addition we concentrate on the physically interesting effect that at
certain critical points in the orbifold moduli spaces additional massless
states appear in the string spectrum leading to logarithmic singularities in
the threshold functions. We discuss this effect for the gauge coupling
threshold corrections; here the appearance of additional massless states is
directly related to the Higgs effect in string theory. In addition the
singularities in the threshold functions are relevant for the loop corrections
to the gravitational coupling constants.
