One approach to computing integrals over Hilbert schemes of points on surfaces (and other moduli spaces of sheaves on surfaces) is to reduce to the special case when the surface in question is $C^2$. \\ \\ I'll explain how to use the (K-theoretic) Donaldson-Thomas theory of threefolds to deduce identities for holomorphic Euler characteristics of tautological bundles over the Hilbert scheme of points on $C^2$. I'll also explain how these identities control the behavior of such Euler characteristics over Hilbert schemes of points on general surfaces.