Partition-theoretic formulas for arithmetic densities

If $\gcd(r,t)=1$, then a theorem of Alladi offers the M\"obius sum identity $$-\sum_{\substack{ n \geq 2 \\ p_{\rm{min}}(n) \equiv r \pmod{t}}} \mu(n)n^{-1}= \frac{1}{\varphi(t)}.$$ Here $p_{\rm{min}}(n)$ is the smallest prime divisor of $n$. The right-hand side represents the proportion of primes in a fixed arithmetic progression modulo $t$. Locus generalized this to Chebotarev densities for Galois extensions. Answering a question of Alladi, we obtain analogs of these results to arithmetic densities of subsets of positive integers using $q$-series and integer partitions. For suitable subsets $\S$ of the positive integers with density $d_{\S}$, we prove that $$- \lim_{q \to 1} \sum_{\substack{ \lambda \in \mathcal{P} \\ \rm{sm}(\lambda) \in \S}} \mu_{\mathcal{P}} (\lambda)q^{\vert \lambda \vert} = d_{\S},$$ where the sum is taken over integer partitions $\lambda$, $\mu_{\mathcal{P}}(\lambda)$ is a partition-theoretic M\"obius function, $\vert \lambda \vert$ is the size of partition $\lambda$, and $\rm{sm}(\lambda)$ is the smallest part of $\lambda$. In particular, we obtain partition-theoretic formulas for even powers of $\pi$ when considering power-free integers.