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  "body": "\n## Summary\n\nThe logarithmic integral is $\\operatorname{li}(x) = \\int_0^x \\frac{dt}{\\ln t}$, taken as a principal value at $t = 1$.\nBy the [[prime number theorem|0EHBY5Y84N]], it is asymptotic to, and an excellent approximation of, the [[prime-counting function|0RQSNVTFVZ]] $\\pi(x)$.\nThe offset version $\\operatorname{Li}(x) = \\int_2^x \\frac{dt}{\\ln t}$ avoids the singularity at $1$.\nIt is the [[exponential integral|EV6ZDQ0NDY]] composed with the logarithm: $\\operatorname{li}(x) = \\operatorname{Ei}(\\ln x)$.\n\n## Explanations\n\n### L1 human\n\nNear a large number $t$, about one number in $\\ln t$ is prime.\nAdding up these densities from $2$ to $x$ gives the integral $\\int_2^x \\frac{dt}{\\ln t}$.\nSo the logarithmic integral predicts how many primes there are up to $x$.\nFor $x = 10^6$ the prediction is about $78\\,627$, while the true count is $78\\,498$.\n",
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