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  "body": "\n## Definition\n\nFor real $x$ write $e(x) = e^{2\\pi i x}$ and let $\\norm{x}$ be the distance from $x$ to the nearest integer.\nLet $M$ be an integer, $N \\ge 1$ an integer and $(a_n)_{M \u003c n \\le M+N}$ complex numbers, and let\n$$S(\\alpha) = \\sum_{M \u003c n \\le M + N} a_n\\, e(n\\alpha),$$\nan [[exponential sum|97PBMQCPPM]].\nLet $\\delta \\in (0, 1]$, and let $\\alpha_1, \\ldots, \\alpha_R$ be real numbers with $\\norm{\\alpha_r - \\alpha_s} \\ge \\delta$ for all $r \\ne s$.\nThe *large sieve inequality* is\n$$\\sum_{r=1}^{R} \\abs{S(\\alpha_r)}^2 \\le \\left(N - 1 + \\delta^{-1}\\right) \\sum_{M \u003c n \\le M+N} \\abs{a_n}^2.$$\nThe *large sieve* is the method that bounds sifted sets by this inequality.\nIts arithmetic form reads as follows.\nFor each prime $p \\le Q$, let $\\omega(p)$ residue classes modulo $p$ be removed, with $0 \\le \\omega(p) \u003c p$.\nIf $\\mathcal{Z}$ is a set of integers in $(M, M+N]$ that meets none of the removed classes, then\n$$\\#\\mathcal{Z} \\le \\frac{N + Q^2}{L}, \\qquad L = \\sum_{q \\le Q} \\mu(q)^2 \\prod_{p \\mid q} \\frac{\\omega(p)}{p - \\omega(p)},$$\nwhere $\\mu$ is the [[Möbius function|0FANE7P0DY]].\n\n## Remarks\n\nExample: take $N = 1$, so $S(\\alpha) = a_{M+1} e((M+1)\\alpha)$ and $\\abs{S(\\alpha_r)}^2 = \\abs{a_{M+1}}^2$ for each $r$.\nThe $\\delta$-separated points in $[0, 1)$ number at most $\\delta^{-1}$, so the left side is at most $\\delta^{-1} \\abs{a_{M+1}}^2$, which is the right side.\nNon-example: if the separation fails, say $\\alpha_1 = \\cdots = \\alpha_R$, the left side is $R \\abs{S(\\alpha_1)}^2$, which exceeds any fixed bound as $R$ grows.\nThe fractions $a/q$ with $q \\le Q$ and $\\gcd(a, q) = 1$ are $Q^{-2}$-separated modulo $1$, which gives the form\n$$\\sum_{q \\le Q} \\sum_{\\substack{a = 1 \\\\ \\gcd(a,q) = 1}}^{q} \\abs{S(a/q)}^2 \\le (N - 1 + Q^2) \\sum_{n} \\abs{a_n}^2.$$\nThe constant $N - 1 + \\delta^{-1}$ is due to Montgomery–Vaughan and Selberg.\n",
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