# Power partitions and saddle-point method

## TL;DR

This paper introduces a simplified saddle-point method-based proof for the asymptotic behavior of the number of partitions of integers into k-th powers, improving upon previous complex proofs.

## Contribution

It provides a new, more straightforward saddle-point approach to derive the asymptotics of p_k(n), streamlining earlier methods by Wright, Vaughan, and Gafni.

## Key findings

- Simplified proof of asymptotic expansion for p_k(n)
- Enhanced clarity over previous methods
- Potential for broader application to partition problems

## Abstract

For $k\geqslant 1$, denote by $p_k(n)$ the number of partitions of an integer $n$ into $k$-th powers. In this note, we apply the saddle-point method to provide a new proof for the well-known asymptotic expansion of $p_k(n)$. This approach turns out to significantly simplify those of Wright (1934), Vaughan (2015) and Gafni (2016).