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

Abstract

Let $Λ(n)$ 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 \[ S_k(x,y;α) = \sum_{x< n \leq x+y} Λ(n) e\left( n^k α\right) \] when $k\geq 3$ for $α$ 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].