Partition-theoretic formulas for arithmetic densities, II

TL;DR

This paper extends partition-theoretic formulas for arithmetic densities of subsets of positive integers, generalizing previous work and connecting to recent Dirichlet series-based results, by developing a new $q$-series density theory.

Contribution

It introduces a $q$-density framework that generalizes earlier partition-theoretic formulas and provides new density formulas for arbitrary subsets of natural numbers.

Findings

Derived new $q$-series formulas for densities of integer subsets.

Extended partition-theoretic formulas to encompass Wang's Dirichlet series generalization.

Established a foundational $q$-density theory for arithmetic densities.

Abstract

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 $q\to \zeta$ 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 Alladi's 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 Wang's 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.