On $R^4$ threshold corrections in IIB string theory and (p,q) string instantons

We obtain the exact non-perturbative thresholds of $R^4$ terms in IIB string theory compactified to eight and seven dimensions. These thresholds are given by the perturbative tree-level and one-loop results together with the contribution of the D-instantons and of the $(p,q)$-string instantons. The invariance under U-duality is made manifest by rewriting the sum as a non-holomorphic modular function of the corresponding discrete U-duality group. In the eight-dimensional case, the threshold is the sum of a order-1 Eisenstein series for $SL(2,Z)$ and a order-3/2 Eisenstein series for $SL(3,Z)$. The seven-dimensional result is given by the order-3/2 Eisenstein series for $SL(5,Z)$. We also conjecture formulae for the non-perturbative thresholds in lower dimensional compactifications and discuss the relation with M-theory.