Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in $\mathcal{ L}\subset\mathbb{ N}$ ($\gcd(\mathcal{ L})=1$) and good analytic properties of the corresponding zeta function, generalizing work of Meinardus. In this paper, we extend their work to prove asymptotic formulas if $\mathcal{ L}$ is a multiset of integers and the zeta function has multiple poles. In particular, our results imply an asymptotic formula for the number of irreducible representations of degree $n$ of $\mathfrak{ so}{ (5)}$. We also study the Witten zeta function $\zeta_{ \mathfrak{ so}{ (5)}}$, which is of independent interest.
Categories: math.NT