= Prime exponential-sum estimate from Vaughan identity
{title2=$|\sum_{n\leq N}\Lambda(n)e(\alpha n)|\ll(Nq^{-1/2}+N^{4/5}+(Nq)^{1/2})\log^4(2N)$}
For reduced $a/q$ with $|\alpha-a/q|\leq q^{-2}$ and $1\leq q\leq N$, the displayed bound follows by splitting <Vaughan identity> with parameters $U=V=N^{2/5}$. <Type I sums> use short geometric-sum bounds; <type II sums> use the <Cauchy-Schwarz inequality> and rational spacing to control bilinear cancellation. A <badly approximable number> has a Dirichlet approximant with $q\asymp N^{1/2}$, yielding $O(N^{4/5}\log^4N)$ and hence $O(N^{9/10})$.
// Target: analytic-number-theory.bigb
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