= Prime-denominator large sieve
When $N\leq P$,
$$
\sum_{\substack{P\leq p\leq2P\\p\text{ prime}}}\sum_{a=1}^{p-1}\left|\sum_{M<n\leq M+N}a_ne(an/p)\right|^2\ll\frac{P^2}{\log P}\sum|a_n|^2.
$$
An arc of length $1/P$ contains at most three points of each grid of denominator $p\leq2P$. The <Chebyshev estimate> gives $O(P/\log P)$ such <primes>. Apply the <local-multiplicity large sieve>. Using only the separation of distinct <reduced fractions> gives the weaker $O(P^2)\sum|a_n|^2$.
Back to article page
