Power Partitions

TL;DR

This paper derives an asymptotic formula for the number of partitions of an integer into k-th powers and provides bounds on related exponential sums, advancing understanding of partition functions.

Contribution

It offers the first rigorous asymptotic formula for p^k(n) and bounds on exponential sums, confirming Hardy and Ramanujan's claims with proof.

Findings

Asymptotic formula for p^k(n) derived

Formula for the difference p^k(n+1)-p^k(n) provided

Non-trivial bounds on exponential sums established

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 $\sum_{m=1}^q e(\frac{am^k}{q})$.