PARTITIONS INTO PRIME POWERS

For a subset A ⊂ N , letp A ( n )denote the restricted partition function which counts partitions of n with all parts lying in A . In this paper, we use a variation of the Hardy–Littlewood circle method to provide an asymptotic formula forp A ( n ) , where 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 analyzep A ( n )for a given set A .