  This paper explores the moduli-dependent coefficients of higher derivative
interactions that appear in the low-energy expansion of the four-graviton
amplitude of maximally supersymmetric string theory compactified on a d-torus.
These automorphic functions are determined for terms up to order D^6R^4 and
various values of d by imposing a variety of consistency conditions. They
satisfy Laplace eigenvalue equations with or without source terms, whose
solutions are given in terms of Eisenstein series, or more general automorphic
functions, for certain parabolic subgroups of the relevant U-duality groups.
The ultraviolet divergences of the corresponding supergravity field theory
limits are encoded in various logarithms, although the string theory
expressions are finite. This analysis includes intriguing representations of
SL(d) and SO(d,d) Eisenstein series in terms of toroidally compactified one and
two-loop string and supergravity amplitudes.
