Eisenstein series in string theory

We discuss the relevance of Eisenstein series for representing certain G (Z )-invariant string theory amplitudes which receive corrections from BPS sta tes only. The Eisenstein series are constructed using G(Z)-invariant mass f ormulae and are manifestly invariant modular functions on the symmetric spa ce 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 s tring theory. We also obtain their non-perturbative extensions in terms of the Eisenstein series Elf the U-duality group Ed+1(d+1)(Z).