  Let $\mathrm{Hilb}_nS$ be the Hilbert scheme of $n$ points on a smooth
projective surface $S$. To a class $\alpha\in K^0(S)$ correspond a tautological
vector bundle $\alpha^{[n]}$ on $\mathrm{Hilb}_nS$ and line bundle
$L_{(n)}\otimes E^{\otimes r}$ with $L=\det(\alpha)$, $r=\mathrm{rk}(\alpha)$.
In this paper we give closed formulas for the generating functions for the
Segre classes $\int_{\mathrm{Hilb}_nS} s(\alpha^{[n]})$, and the Verlinde
numbers $\chi(\mathrm{Hilb}_nS,L_{(n)}\otimes E^{\otimes r})$, for any surface
$S$ and any class $\alpha\in K^0(S)$. In fact we determine a more general
generating function for $K$-theoretic invariants of Hilbert schemes of points,
which contains the formulas for Segre and Verlinde numbers as specializations.
We prove these formulas in case $K_S^2=0$. Without assuming the condition
$K_S^2=0$, we show the Segre-Verlinde conjecture of Johnson and
Marian-Oprea-Pandharipande, which relates the Segre and Verlinde generating
series by an explicit change of variables.
