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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be a finite set of integers, let $\\mathcal{P}$ be a set of primes and let $z \\ge 2$ be real.\nLet $P(z)$ be the product of the primes $p \\in \\mathcal{P}$ with $p \u003c z$.\nFor $d \\ge 1$ let $\\mathcal{A}_d = \\set{a \\in \\mathcal{A} : d \\mid a}$.\nThe sifting function is\n$$S(\\mathcal{A}, \\mathcal{P}, z) = \\#\\set{a \\in \\mathcal{A} : \\gcd(a, P(z)) = 1}.$$\nLet $\\mu$ be the [[Möbius function|0FANE7P0DY]] and $\\omega(d)$ the [[number of distinct prime factors|08RQ62VAVA]] of $d$.\nThe *Brun sieve* in its simplest form, Brun's pure sieve, uses the truncated sums\n$$S_r = \\sum_{\\substack{d \\mid P(z) \\\\ \\omega(d) \\le r}} \\mu(d)\\, \\#\\mathcal{A}_d,$$\ntogether with the inequalities, valid for every integer $r \\ge 0$,\n$$S_{2r+1} \\le S(\\mathcal{A}, \\mathcal{P}, z) \\le S_{2r}.$$\n\n## Remarks\n\nExample: let $\\mathcal{A} = \\set{1, \\ldots, 10}$, let $\\mathcal{P}$ be all primes and $z = 4$, so $P(z) = 6$.\nThen $S(\\mathcal{A}, \\mathcal{P}, 4) = \\#\\set{1, 5, 7} = 3$.\nHere $S_0 = 10$ and $S_1 = 10 - 5 - 3 = 2$, so $S_1 \\le 3 \\le S_0$; and $S_2 = 2 + \\#\\mathcal{A}_6 = 3$.\nThe inequalities are the Bonferroni inequalities: for $m \\ge 1$, the partial sums $\\sum_{j \\le k} (-1)^j \\binom{m}{j}$ are $\\ge 0$ for even $k$ and $\\le 0$ for odd $k$.\nNon-example: the full Legendre sum $S_r$ with $r \\ge \\omega(P(z))$ is exact, but it is not a truncation.\nIt has $2^{\\omega(P(z))}$ terms, and the errors in approximating each $\\#\\mathcal{A}_d$ add up.\nBrun's later, more refined sieve replaces the condition $\\omega(d) \\le r$ by conditions on the sizes of the prime factors of $d$.\n",
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