On sums of powers of almost equal primes

Abstract

Let $k\ge 2$ and $s$ be positive integers, and let $n$ be a large positive integer subject to certain local conditions. We prove that if $s\ge k^2+k+1$ and $\theta >31/40$, then $n$ can be expressed as a sum $p_1^k+\cdots +p_s^k$, where $p_1,\dots ,p_s$ are primes with $|p_j-(n/s{)}^{1/k}|\le n^{\theta /k}$. This improves on earlier work by Wei and Wooley and by Huang who proved similar theorems when $\theta >19/24$.

References in corpus (1)

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