Exponential sums over primes in short intervals and an application to the Waring--Goldbach problem

TL;DR

This paper refines estimates for exponential sums over primes in short intervals for higher powers and applies these results to improve the understanding of representing integers as sums of prime powers in the Waring--Goldbach problem.

Contribution

It provides improved bounds on exponential sums over primes in short intervals for k ≥ 3 and applies these to enhance results on the Waring--Goldbach problem.

Findings

Improved estimates for exponential sums over primes in short intervals.

Enhanced results on representing integers as sums of prime k-th powers.

Application of refined bounds to the Waring--Goldbach problem.

Abstract

Let $\Lambda(n)$ be the von Mangoldt function, $x$ real and $2\leq y \leq x$. This paper improves the estimate on the exponential sum over primes in short intervals \[ S_k(x,y;\alpha) = \sum_{x< n \leq x+y} \Lambda(n) e\left( n^k \alpha \right) \] when $k\geq 3$ for $\alpha$ in the minor arcs. And then combined with the Hardy--Littlewood circle method, this enables us to investigate the Waring--Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers, which improves the results in Wei and Wooley [12].