Exponential sums over primes

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