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  "body": "\n## Definition\n\nLet $\\ln$ be the [[natural logarithm|04DXAFN2ES]].\nFor a real number $x \u003e 0$ with $x \\ne 1$, the *logarithmic integral* is\n$$\\operatorname{li}(x) = \\int_0^x \\frac{dt}{\\ln t},$$\nwhere for $x \u003e 1$ the integral is the [[Cauchy principal value|0SYR10HD4R]] at $t = 1$:\n$$\\operatorname{li}(x) = \\lim_{\\varepsilon \\to 0^+} \\left( \\int_0^{1 - \\varepsilon} \\frac{dt}{\\ln t} + \\int_{1 + \\varepsilon}^x \\frac{dt}{\\ln t} \\right).$$\nThe *offset logarithmic integral* is $\\operatorname{Li}(x) = \\operatorname{li}(x) - \\operatorname{li}(2) = \\int_2^x \\frac{dt}{\\ln t}$ for $x \\ge 2$.\n\n## Remarks\n\nThe integrand tends to $0$ as $t \\to 0^+$, so there is no problem at the lower end.\n\nExample: the substitution $t = e^u$ gives $\\operatorname{li}(x) = \\operatorname{Ei}(\\ln x)$, where $\\operatorname{Ei}$ is the [[exponential integral|EV6ZDQ0NDY]].\nThe function $\\operatorname{li}$ has exactly one positive zero, at $x = \\mu \\approx 1.4513$, the Ramanujan–Soldner constant, and $\\operatorname{li}(2) \\approx 1.045$.\n\nNon-example: without the principal value the integral diverges for $x \u003e 1$, because $\\frac{1}{\\ln t} \\sim \\frac{1}{t - 1}$ near $t = 1$ and $\\int \\frac{dt}{\\abs{t - 1}}$ diverges.\n\nThe [[prime number theorem|0EHBY5Y84N]] is equivalent to $\\pi(x) \\sim \\operatorname{li}(x)$, where $\\pi$ is the [[prime-counting function|0RQSNVTFVZ]].\nIntegration by parts gives $\\operatorname{li}(x) \\sim \\frac{x}{\\ln x}$ as $x \\to \\infty$.\n",
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