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  "body": "\n## Statement\n\nLet $\\pi(x)$ be the [[prime-counting function|0XRJB5SSNE]], and let $\\ln$ be the natural logarithm.\nThen\n$$\\lim_{x \\to \\infty} \\frac{\\pi(x)}{x / \\ln x} = 1.$$\n\n## Remarks\n\nWe write this as $\\pi(x) \\sim x / \\ln x$.\nAn equivalent form: the $n$-th prime $p_n$ satisfies $p_n \\sim n \\ln n$.\nThe logarithmic integral $\\operatorname{li}(x) = \\int_2^x \\frac{dt}{\\ln t}$ gives a better approximation of $\\pi(x)$, with the same limit.\nFor example, $\\pi(10^6) = 78498$ and $10^6 / \\ln 10^6 \\approx 72382$.\n\n## History\n\nLegendre and Gauss conjectured the result from tables of primes, at the end of the 18th century.\nHadamard and de la Vallée Poussin proved it independently in 1896, with methods of complex analysis. Selberg and Erdős found elementary proofs in 1949.\n\n## Explanations\n\n### L1 human\n\nNear a large number $x$, about one number in $\\ln x$ is prime.\nFor numbers with $100$ digits, $\\ln x \\approx 230$, so about one number in $230$ is prime.\nThe theorem makes this rough rule precise.\n",
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