  We compute the generating series for the intersection pairings between the
total Chern classes of the tangent bundles of the Hilbert schemes of points on
a smooth projective surface and the Chern characters of tautological bundles
over these Hilbert schemes. Modulo the lower weight term, we verify Okounkov's
conjecture [Oko] connecting these Hilbert schemes and multiple $q$-zeta values.
In addition, this conjecture is completely proved when the surface is abelian.
We also determine some universal constants in the sense of Boissi\' ere and
Nieper-Wisskirchen [Boi, BN] regarding the total Chern classes of the tangent
bundles of these Hilbert schemes. The main approach of this paper is to use the
set-up of Carlsson and Okounkov outlined in [Car, CO] and the structure of the
Chern character operators proved in [LQW2].
