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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 [ 13 ] 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 Boissière and Nieper-Wisskirchen [ 1 ,  2 ] regarding the total Chern classes of the tangent bundles of these Hilbert schemes. The main approach of this article is to use the set-up of Carlsson and Okounkov outlined in Carlsson [ 5 ,  6 ] and the structure of the Chern character operators proved in Li, Qin, and Wang [ 10 ].

On Okounkov’s Conjecture Connecting Hilbert Schemes of Points and Multiple $q$ -Zeta Values