Partition-theoretic formulas for arithmetic densities

Abstract

If $\gcd (r,t)=1$, then a theorem of Alladi offers the Möbius sum identity $$-\sum _{\begin{array}{c}n\ge 2\\ p_{\min }(n)\equiv r(\mathrm{mod}t)\end{array}}\mu (n)n^{-1}=\frac{1}{\phi (t)}.$$ Here $p_{\min }(n)$ 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 \[- \lim_{q \to 1} \sum_{\substack{ λ\in \mathcal{P} \\ \rm{sm}(λ) \in §}} μ_{\mathcal{P}} (λ)q^{\vert λ\vert} = d_§,\] where the sum is taken over integer partitions $\lambda$, ${\mu }_{\mathcal{P}}(\lambda )$ is a partition-theoretic Möbius function, $|\lambda |$ is the size of partition $\lambda$, and $\mathrm{sm}(\lambda )$ is the smallest part of $\lambda$. In particular, we obtain partition-theoretic formulas for even powers of $\pi$ when considering power-free integers.

The second version has been accepted for publication in Proceedings of Number Theory in Honor of Krishna Alladi's 60th Birthday, Springer