Bounds on the Möbius-signed partition numbers

For \(n \in \mathbb {N}\) let \(\Pi [n]\) denote the set of partitions of n, i.e., the set of positive integer tuples \((x_1,x_2,\ldots ,x_k)\) such that \(x_1 \ge x_2 \ge \ldots \ge x_k\) and \(x_1 + x_2 + \cdots + x_k = n\). Fixing \(f:\mathbb {N}\rightarrow \{0,\pm 1\}\), for \(\pi = (x_1,x_2,\ldots ,x_k) \in \Pi [n]\) let \(f(\pi ) := f(x_1)f(x_2)\cdots f(x_k)\). In this way we define the signed partition numbers

$$\begin{aligned} p(n,f) = \sum _{\pi \in \Pi [n]} f(\pi ). \end{aligned}$$

Following work of Vaughan and Gafni on partitions into primes and prime powers, we derive asymptotic formulae for \(p(n,\mu )\) and \(p(n,\lambda )\), where \(\mu \) and \(\lambda \) 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.