Discorrelation Between Primes in Short Intervals and Polynomial Phases

Abstract

Abstract Let $H = N^{\theta }, \theta> 2/3$ and $k \geq 1$. We obtain estimates for the following exponential sum over primes in short intervals: $$\begin{equation*} \sum_{N < n \leq N+H} \Lambda(n) \mathrm e(g(n)), \end{equation*}$$where $g$ is a polynomial of degree $k$. As a consequence of this in the special case $g(n) = \alpha n^k$, we deduce a short interval version of the Waring–Goldbach problem.