Exponential sums over primes

Abstract

Let $\alpha =a/q+\epsilon$ with $(a,q)=1$, $q\le N^{1/2}$ and $|\epsilon |\le 1/(q{N}^{1/2})$, and let $B:=\max (q,qN|\epsilon |)$. We show that \[ \Bigl|\sum_{n<N}Λ(n)e(nα)\Bigr|\le N^{o(1)}\Bigl(\frac{N}{B^{1/2}}+N^{19/24}\Bigr). \] This improves on the classical bound of Vinogradov from 1937, which has $N^{4/5}$ in place of $N^{19/24}$.

17 pages