Discorrelation between primes in short intervals and polynomial phases

Abstract

Let $H=N^{\theta },\theta >2/3$ and $k\ge 1$. We obtain estimates for the following exponential sum over primes in short intervals: \[ \sum_{N < n \leq N+H} Λ(n) e(g(n)), \] 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.

20 pages

References in corpus (1)

• Strong orthogonality between the Mobius function and nonlinear exponential functions in short intervals