Combinatorial formulas for arithmetic density

Abstract

Let $d_S$ denote the arithmetic density of a subset $S\subseteq \mathbb{N}$. We derive a power series in $q\in \mathbb{C}$, $|q|<1$, with coëfficients related to integer partitions and integer compositions, that yields $1/d_S$ in the limit as $q\to 1$ radially.

5 pages