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  "body": "\n## Summary\n\nThe large sieve is a method of analytic number theory.\nIt applies when a set of integers is sifted by many residue classes modulo each prime.\nIts core is an inequality for exponential sums: a trigonometric polynomial cannot be large at many well-spaced points at once.\nIn arithmetic form it bounds the size of a set of integers that misses many residue classes modulo many primes.\nIt is a key tool in the proof of the [[Bombieri–Vinogradov theorem|B3JMY54GJF]].\n\n## Explanations\n\n### L1 human\n\nClassical sieves remove a few residue classes modulo each prime, for example the class $0$.\nThe large sieve handles the case where a large fraction of the classes is removed.\nFor example, the perfect squares fall into only $(p+1)/2$ of the $p$ residue classes modulo an odd prime $p$.\nIts engine is a kind of uncertainty principle: a sum of waves $e^{2\\pi i n \\alpha}$ with $n$ in a short interval cannot be big at many points $\\alpha$ that are far apart.\nA set of integers that avoids many residue classes would make such a sum big at many fractions $a/q$, so such a set must be small.\n",
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