= Large sieve upper bound for sifted intervals
{title2=$|S|\ll(H+Q^2)/\sum_{d\le Q}\mu^2(d)\prod_{p\mid d}(p-1)^{-1}$}
For an interval set avoiding one residue at each sieving <prime>, the <Ramanujan sum> at the forbidden <Chinese remainder theorem> residue gives a linear combination of Fourier samples equal to $\mu(d)|S|$. <Cauchy-Schwarz inequality> gives sample energy at least $|S|^2/\varphi(d)$ for each <squarefree> modulus. Summing these energies and applying the <analytic large sieve inequality> gives the bound. If <primes> dividing $q$ are excluded, restrict the denominator to $(d,q)=1$; it is uniformly at least a constant times $(\varphi(q)/q)\log(2Q)$.
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