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  "body": "\n## Summary\n\nThe Selberg sieve is an upper bound sieve of Atle Selberg.\nIt replaces the indicator of \"no prime factor below $z$\" by the square of a sum of real weights.\nIt then chooses the weights to minimise the main term, which is a quadratic form.\nIt gives sharp upper bounds with a simple remainder term, for example the Brun–Titchmarsh bound for primes in arithmetic progressions.\nIt is also called the $\\Lambda^2$ sieve.\n\n## Explanations\n\n### L1 human\n\nA sieve wants to count the numbers in a list that are not divisible by any small prime.\nSelberg's trick is to use the fact that a square is never negative.\nHe builds, for each number, a square of a sum of weights that equals $1$ when the number survives and is $\\ge 0$ otherwise.\nSumming these squares gives an upper bound for the count, and he then picks the weights that make this bound as small as possible.\nThe method gives only upper bounds, but these are often close to the truth.\n",
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