Prime exponential-sum estimate from Vaughan identity ($|{\sum }_{n\le N}\Lambda (n)e(\alpha n)|\ll (N{q}^{-1/2}+N^{4/5}+(Nq{)}^{1/2}){\log }^4(2N)$)

For reduced $a/q$ with $|\alpha -a/q|\le q^{-2}$ and $1\le q\le N$, 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 $O(N^{4/5}{\log }^{4}N)$ and hence $O(N^{9/10})$.

// Target: analytic-number-theory.bigb