Power Partitions

Abstract

In 1918, Hardy and Ramanujan published a seminal paper which included an asymptotic formula for the partition function. In their paper, they also claim without proof an asymptotic equivalence for $p^k(n)$, the number of partitions of a number $n$ into $k$-th powers. In this paper, we provide an asymptotic formula for $p^k(n)$, using the Hardy-Littlewood Circle Method. We also provide a formula for the difference function $p^k(n+1)-p^k(n)$. As a necessary step in the proof, we obtain a non-trivial bound on exponential sums of the form $\sum_{m=1}^q e(\frac{am^k}{q})$.