  We study the coupling constant renormalization of gauge theories with an
infinite multiplet of fermions, using the zeta function method to make sense of
the infinite sums over fermions. If the gauge group K is the maximal compact
subgroup of a simple non-compact group G, such infinite multiplets can arise
naturally, as reductions of discrete series unitary representations of G. The
example K=U(1) and G=SU(1,1) will be studied in detail. Surprisingly, there are
abelian gauge theories which are asymptotically free; and others that are UV
finite.
