  In earlier work generalizing a 1977 theorem of Alladi, the authors proved a
partition-theoretic formula to compute arithmetic densities of certain subsets
of the positive integers $\mathbb N$ as limiting values of $q$-series as $q\to
\zeta$ a root of unity (instead of using the usual Dirichlet series to compute
densities), replacing multiplicative structures of $\mathbb N$ by analogous
structures in the integer partitions $\mathcal P$. In recent work, Wang obtains
a wide generalization of Alladi's original theorem, in which arithmetic
densities of subsets of prime numbers are computed as values of Dirichlet
series arising from Dirichlet convolutions. Here the authors prove that Wang's
extension has a partition-theoretic analogue as well, yielding new $q$-series
density formulas for any subset of $\mathbb N$. To do so, we outline a theory
of $q$-series density calculations from first principles, based on a statistic
we call the "$q$-density" of a given subset. This theory in turn yields
infinite families of further formulas for arithmetic densities.
