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  "body": "\n## Definition\n\nLet $x$ be a real number.\nThe *first Chebyshev function* is\n$$\\vartheta(x) = \\sum_{p \\le x} \\log p,$$\nwhere $p$ runs over the [[prime numbers|01826KMG2H]] at most $x$ and $\\log$ is the [[natural logarithm|04DXAFN2ES]].\nThe *second Chebyshev function* is\n$$\\psi(x) = \\sum_{\\substack{p, k \\\\ p^k \\le x}} \\log p = \\sum_{1 \\le n \\le x} \\Lambda(n).$$\nHere the first sum runs over the pairs of a prime $p$ and an integer $k \\ge 1$.\nThe symbol $\\Lambda$ denotes the [[von Mangoldt function|0NK8JTE1NN]].\n\n## Remarks\n\nExample: $\\vartheta(10) = \\log 2 + \\log 3 + \\log 5 + \\log 7 = \\log 210$.\nThe prime powers up to $10$ are $2, 4, 8, 3, 9, 5, 7$, so $\\psi(10) = \\log(2^3 \\cdot 3^2 \\cdot 5 \\cdot 7) = \\log 2520$.\nIn general $\\psi(x) = \\log \\operatorname{lcm}(1, 2, \\dots, \\floor{x})$ for $x \\ge 1$.\nNon-example: $\\sum_{n \\le 10} \\log n = \\log 10!$ is not $\\psi(10)$, since it also has terms such as $\\log 6$ for $n$ that are not prime powers.\nThe two functions are related by $\\psi(x) = \\sum_{k \\ge 1} \\vartheta(x^{1/k})$, and $\\psi(x) - \\vartheta(x) = O(\\sqrt{x} \\log x)$.\nBoth are $0$ for $x \u003c 2$.\n",
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