  We discuss the relevance of Eisenstein series for representing certain
G(Z)-invariant string theory amplitudes which receive corrections from BPS
states only. The Eisenstein series are constructed using G(Z)-invariant mass
formulae and are manifestly invariant modular functions on the symmetric space
K\G(R) of non-compact type, with K the maximal compact subgroup of G(R). In
particular, we show how Eisenstein series of the T-duality group SO(d,d,Z) can
be used to represent one- and g-loop amplitudes in compactified string theory.
We also obtain their non-perturbative extensions in terms of the Eisenstein
series of the U-duality group E_{d+1(d+1)}(Z).

High Energy Physics - Theory