James Maynard

Exponential sums over primes

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