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We compute the coefficients of the polynomials $C_n(q)$ defined by theequation \begin{equation*} 1 + \sum_{n\geq 1} \, \frac{C_n(q)}{q^n} \, t^n =\prod_{i\geq 1}\, \frac{(1-t^i)^2}{1-(q+q^{-1})t^i + t^{2i}} \, .\end{equation*} As an application we obtain an explicit formula for the zetafunction of the Hilbert scheme of $n$ points on a two-dimensional torus andshow that this zeta function satisfies a remarkable functional equation. Thepolynomials $C_n(q)$ are divisible by $(q-1)^2$. We also compute thecoefficients of the polynomials $P_n(q) = C_n(q)/(q-1)^2$: each coefficientcounts the divisors of $n$ in a certain interval; it is thus a non-negativeinteger. Finally we give arithmetical interpretations for the values of$C_n(q)$ and of $P_n(q)$ at $q = -1$ and at roots of unity of order $3$, $4$,$6$.

Complete determination of the zeta function of the Hilbert scheme of $n$ points on a two-dimensional torus