Hilbert schemes of points on surfaces and multiple q-zeta values

For a line bundle L on a smooth projective surface X and nonnegative integers k(1), ... , k(N), Okounkov [19] introduced the reduced generating series < ch(k1)(L) ... ch(kN)(L)>' for the intersection numbers among the Chern characters of the tautological bundles over the Hilbert schemes of points on X and the total Chern classes of the tangent bundles of these Hilbert schemes, and conjectured that they are multiple q-zeta values of weight at most Sigma(N)(i=1) (k(i)+2). The second-named author further conjectured in [22] that these reduced generating series are quasi-modular forms if the canonical divisor of X is numerically trivial. In this paper, we verify these two conjectures for < ch(2)(L)>'. The main approaches are to apply the procedure laid out in [23] and to establish various identities for multiple q-zeta values and quasi-modular forms.