Partition-Theoretic Formulas for Arithmetic Densities

Abstract

If $\gcd (r,t)=1$, then Alladi proved the Möbius sum identity

$$ -\sum _{\begin{array}{c} n \ge 2 \\ p_\mathrm{{min}}(n) \equiv r \pmod {t} \end{array}} \mu (n)n^{-1}= \frac{1}{\varphi (t)}. $$

Here $p_\mathrm{{min}}(n)$ 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 ${\mathcal S}$ of the positive integers with density $d_{{\mathcal S}}$, we prove that

$$\begin{aligned} - \lim _{q \rightarrow 1} \sum _{\begin{array}{c} \lambda \in \mathscr {P} \\ \mathrm {sm}(\lambda ) \in {\mathcal S} \end{array}} \mu _{\mathscr {P}} (\lambda )q^{\vert \lambda \vert } = d_{{\mathcal S}}, \end{aligned}$$

where the sum is taken over integer partitions $\lambda$, $\mu _{\mathscr {P}}(\lambda )$ is a partition-theoretic Möbius function, $\vert \lambda \vert$ 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.

In celebration of Krishnaswami Alladi’s 60th birthday

This research was partially supported by NSF grant DMS 1601306. The first author thanks the support of the Asa Griggs Candler Fund, and the second author thanks the support of an Emory University Woodruff Fellowship.