On the Waring-Goldbach problem on average

TL;DR

This paper establishes an asymptotic formula for the average number of representations of integers as sums of prime powers within short intervals, advancing understanding of the Waring-Goldbach problem on average.

Contribution

It provides the first asymptotic formula for the average number of prime power representations in short intervals for the Waring-Goldbach problem.

Findings

Asymptotic formula proven for average representations

Results apply to short intervals

Advances understanding of prime power sums

Abstract

Let $s$, $\ell$ be two integers such that $2\le s\le \ell-1$, $\ell\ge 3$. We prove that a suitable asymptotic formula for the average number of representations of integers $n=\sum_{i=1}^{s} p_{i}^{\ell}$, where $p_i$, $i=1,\dotsc,s$, are prime numbers, holds in short intervals.