On the mean average of integer partition as the sum of powers

TL;DR

This paper investigates the average number of ways to express integers as sums of positive powers, focusing on the sum of the representation counts up to n, providing insights into the distribution of such representations.

Contribution

It introduces an analysis of the mean average of the representation function for sums of powers, extending understanding of integer partitions into powers.

Findings

Derived formulas for the mean average of representation counts

Identified asymptotic behavior of the sum of representations

Provided bounds and estimates for the number of representations

Abstract

This paper is concerned with the function $r_{k,s}(n)$, the number of (ordered) representations of $n$ as the sum of $s$ positive $k$-th powers, where integers $k,s\ge 2$. We examine the mean average of the function, or equivalently, \begin{equation*} \sum_{m=1}^n r_{k,s}(m). \end{equation*}