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 $qto zeta$ 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 Alladis 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 Wangs 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.
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