= Major-arc value of the prime exponential sum
{title2=$\sum_{n\le N}\Lambda(n)e(an/q)=N\mu(q)/\varphi(q)+O_{A,B}(N\log^{-B}N)$}
For $(a,q)=1$ and $q\le\log^A N$, apply the <Siegel–Walfisz theorem> to every reduced residue class with error exponent $A+B+2$. The main terms sum to $N c_q(a)/\varphi(q)$, where the <Ramanujan sum> satisfies $c_q(a)=\mu(q)$ for coprime $a,q$. Nonunits with nonzero <Von Mangoldt function> are powers of primes dividing $q$ and contribute at most $\omega(q)\log N$, which is absorbed by the error. For arbitrary $a$, the correct main term is $N c_q(a)/\varphi(q)$; the reduced numerator hypothesis cannot be omitted.
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