Bounds on the Möbius-signed partition numbers

Abstract

For $n\in \mathbb{N}$ let $\hat{I}[n]$ denote the set of partitions of $n$, i.e., the set of positive integer tuples $(x_1,x_2,\dots ,x_k)$ such that $x_1\ge x_2\ge \cdots \ge x_k$ and $x_1+x_2+\cdots +x_k=n$. Fixing $f:\mathbb{N}\to \{0,\pm 1\}$, for $\ddot{I}€=(x_1,x_2,\dots ,x_k)\in \hat{I}[n]$ let $f(\ddot{I}€):=f(x_1)f(x_2)\cdots f(x_k)$. In this way we define the {signed partition numbers} \[ p(n,f) = \sum_{π\inÎ [n]} f(π). \] Following work of Vaughan and Gafni on partitions into primes and prime powers, we derive asymptotic formulae for quantities $p(n,\hat{I}¼)$ and $p(n,\hat{I}»)$, where $\hat{I}¼$ and $\hat{I}»$ denote the Möbius and Liouville functions from prime number theory, respectively. In addition we discuss how quantities $p(n,f)$ generalize the classical notion of restricted partitions.

Updated to reflect grammatical/typographical edits made during publication; statement and proof of Proposition 5.4 have been greatly simplified, allowing for the deletion of the appendix; 35 pages, 1 figure