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

We obtain the full non-perturbative thresholds of $R^4$ terms in IIB string theory compactified to eight and seven dimensions. They are given by the perturbative results plus a sum over D-instantons and (p,q) string instantons. The thresholds are shown to be U-duality invariant. They are given by a weight-3/2 non holomorphic form of SL(3,Z) and a weight-1 form of SL(2,Z) in the eight-dimensional case. In seven dimensions they are given by a weight-3/2 form of SL(5,Z). We conjecture also formulae for the non-perturbative thresholds in lower dimensional compactifications.