On the average number of representations of an integer as a sum of like prime powers

TL;DR

This paper studies the average number of ways to express integers as sums of prime powers, extending previous work to include sums of exactly k prime powers, revealing new insights into their distribution.

Contribution

It generalizes existing results on sums of prime squares and cubes to sums of k prime powers, providing new theoretical bounds and methods.

Findings

Extended results to sums of k prime powers

Derived new bounds for average representations

Applied techniques to broader classes of prime power sums

Abstract

We investigate the average number of representations of a positive integer as the sum of $k + 1$ perfect $k$-th powers of primes. We extend recent results of Languasco and the last Author, which dealt with the case $k = 2$ [6] and $k = 3$ [5] respectively. We use the same technique to study the corresponding problem for sums of just $k$ perfect $k$-th powers of primes.