Bounds on the Möbius-signed partition numbers

For \(n \in \mathbb {N}\) $n\in \mathbb{N}$ let \(\Pi [n]\) $\mathrm{\Pi }[n]$ denote the set of partitions of n, i.e., the set of positive integer tuples \((x_1,x_2,\ldots ,x_k)\) $(x_1,x_2,\dots ,x_k)$ such that \(x_1 \ge x_2 \ge \ldots \ge x_k\) $x_1\ge x_2\ge \dots \ge x_k$ and \(x_1 + x_2 + \cdots + x_k = n\) $x_1+x_2+\cdots +x_k=n$ . Fixing \(f:\mathbb {N}\rightarrow \{0,\pm 1\}\) $f:\mathbb{N}\to \{0,\pm 1\}$ , for \(\pi = (x_1,x_2,\ldots ,x_k) \in \Pi [n]\) $\pi =(x_1,x_2,\dots ,x_k)\in \mathrm{\Pi }[n]$ let \(f(\pi ) := f(x_1)f(x_2)\cdots f(x_k)\) $f(\pi ):=f(x_1)f(x_2)\cdots f(x_k)$ . In this way we define the signed partition numbers \(\begin{aligned} p(n,f) = \sum _{\pi \in \Pi [n]} f(\pi ). \end{aligned}\) $$\begin{array}{c}\hfill p(n,f)=\sum _{\pi \in \mathrm{\Pi }[n]}f(\pi ).\end{array}$$ Following work of Vaughan and Gafni on partitions into primes and prime powers, we derive asymptotic formulae for \(p(n,\mu )\) $p(n,\mu )$ and \(p(n,\lambda )\) $p(n,\lambda )$ , where \(\mu \) $\mu$ and \(\lambda \) $\lambda$ 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.