  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.
