= Variance form of the large sieve
For coefficients $b_n$ supported on an interval of $H$ consecutive <integers>, set $B=\sum b_n$ and $B_p(a)=\sum_{n\equiv a\pmod p}b_n$. Then
$$
\sum_{p\leq Q}p\sum_{a\bmod p}|B_p(a)-B/p|^2\leq(Q^2+2\pi H)\sum|b_n|^2.
$$
The <orthogonality of roots of unity> gives $p\sum_a|B_p(a)-B/p|^2=\sum_{r=1}^{p-1}|\sum_nb_ne(rn/p)|^2$. The distinct fractions $r/p$ have <circular spacing> at least $Q^{-2}$; now apply the <exponential-sum large sieve>.
Back to article page
