On Sums of Powers of Almost Equal Primes

TL;DR

This paper advances the understanding of the Waring-Goldbach problem by demonstrating that large integers can be expressed as sums of almost equal prime powers, using new Weyl-type estimates for short interval exponential sums.

Contribution

It introduces a novel Weyl-type estimate for exponential sums over short intervals, enabling new results on representing integers as sums of almost equal prime powers.

Findings

Representation of large integers as sums of prime powers within short intervals.

Established bounds for the number of prime powers needed based on the power k.

New techniques for exponential sums with variables in constrained short intervals.

Abstract

We investigate the Waring-Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers. Define $s_k=2k(k-1)$ when $k\ge 3$, and put $s_2=6$. In addition, put $\theta_2=\frac{19}{24}$, $\theta_3=\frac{4}{5}$ and $\theta_k=\frac{5}{6}$ $(k\ge 4)$. Suppose that $n$ satisfies the necessary congruence conditions, and put $X=(n/s)^{1/k}$. We show that whenever $s>s_k$ and $\varepsilon>0$, and $n$ is sufficiently large, then $n$ is represented as the sum of $s$ $k$th powers of prime numbers $p$ with $|p-X|\le X^{\theta_k+\varepsilon}$. This conclusion is based on a new estimate of Weyl-type specific to exponential sums having variables constrained to short intervals.