Partitions into prime powers

Abstract

For a subset $\mathcal A\subset \mathbb N$, let $p_{\mathcal A}(n)$ denote the restricted partition function which counts partitions of $n$ with all parts lying in $\mathcal A$. In this paper, we use a variation of the Hardy-Littlewood circle method to provide an asymptotic formula for $p_{\mathcal A}(n)$, where $\mathcal A$ is the set of $k$-th powers of primes (for fixed $k$). This combines Vaughan's work on partitions into primes with the author's previous result about partitions into $k$-th powers. This new asymptotic formula is an extension of a pattern indicated by several results about restricted partition functions over the past few years. Comparing these results side-by-side, we discuss a general strategy by which one could analyze $p_{\mathcal A}(n )$ for a given set $\mathcal A$.