  We work out the relation between automorphic forms on SO(s+2,2) and gauge
one-loop corrections of heterotic K3 x T^2 string compactifications for the
cases s=0,1. We find that one-loop gauge corrections of any orbifold limit of
K3 can always be expressed by their instanton numbers and generic automorphic
forms.These functions classify also one-loop gauge thresholds of N=1 (0,2)
heterotic compactifications based on toroidal orbifolds T^6/Z_N. We compare
these results with the gauge couplings of M-theory compactified on S^1/Z_2 x
T^6/Z_N using Witten's Calabi-Yau strong coupling expansion.
