Partitions into prime powers

TL;DR

This paper develops an asymptotic formula for the number of partitions of an integer into prime power parts using a variation of the Hardy-Littlewood circle method, extending previous work on prime and power partitions.

Contribution

It introduces a new asymptotic formula for partitions into prime powers, combining Vaughan's prime partition results with earlier power partition findings.

Findings

Provides an asymptotic formula for $p_{ ext{powers of primes}}(n)$

Extends previous results on restricted partition functions

Suggests a general strategy for analyzing partitions into various sets

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 $\mathcal A$.