  In this paper we conduct a general, model-independent analysis of the running
of gauge couplings within closed string theories. Unlike previous discussions
in the literature, our calculations fully respect the underlying modular
invariance of the string and include the contributions from the infinite towers
of string states which are ultimately responsible for many of the properties
for which string theory is famous, including an enhanced degree of finiteness
and UV/IR mixing. In order to perform our calculations, we adopt a formalism
that was recently developed for calculations of the Higgs mass within such
theories, and demonstrate that this formalism can also be applied to
calculations of gauge couplings. In general, this formalism gives rise to an
``on-shell'' effective field theory (EFT) description in which the final
results are expressed in terms of supertraces over the physical string states,
and in which these quantities exhibit an EFT-like ``running'' as a function of
an effective spacetime mass scale. We find, however, that the calculation of
the gauge couplings differs in one deep way from that of the Higgs mass: while
the latter results depend on purely on-shell supertraces, the former results
have a different modular structure which causes them to depend on off-shell
supertraces as well. In some regions of parameter space, our results
demonstrate how certain expected field-theoretic behaviors can emerge from the
highly UV/IR-mixed environment. In other situations, by contrast, our results
give rise to a number of intrinsically stringy behaviors that transcend what
might be expected within an effective field theory approach.
