  A theorem of G\"ottsche establishes a connection between cohomological
invariants of a complex projective surface $S$ and corresponding invariants of
the Hilbert scheme of $n$ points on $S.$ This relationship is encoded in
certain infinite product $q$-series which are essentially modular forms. Here
we make use of the circle method to arrive at exact formulas for certain
specializations of these $q$-series, yielding convergent series for the
signature and Euler characteristic of these Hilbert schemes. We also analyze
the asymptotic and distributional properties of the $q$-series' coefficients.
