  Invariance of Type IIB superstring theory under SL(2,Z) or S-duality implies
dependence on the complex coupling T through real and complex modular forms in
T. Their structure may be understood explicitly in an expansion of superstring
corrections to Einstein's equations of gravity, in powers of derivatives D and
curvature R. The perturbative loop expansion in the string coupling for the
4-string amplitude governs corrections of the form D^{2p} R^4 for all p. We
show that, at two-loop order, the D^6 R^4 term is proportional to the integral
of a modular invariant introduced by Zhang and Kawazumi in number theory and
related to the Faltings delta-invariant studied for genus-two by Bost. The
structure of two-loop superstring amplitudes for p>3 leads to higher
invariants, which generalize Zhang--Kawazumi invariants at genus two. An
explicit formula is derived for the unique higher invariant associated with
order D^8 R^4. In an attempt to compare the prediction for the D^6 R^4
correction from superstring perturbation theory with the one produced by
S-duality and supersymmetry of Type IIB, various reformulations of the
invariant are given. This comparison with string theory leads to a predicted
value for the integral of the Zhang-Kawazumi invariant over the moduli space of
genus-two surfaces.
