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  "body": "\n## Definition\n\nThe von Mangoldt function $\\Lambda$ is the arithmetic function given, for each positive integer $n$, by\n$$\\Lambda(n) = \\begin{cases} \\log p \u0026 \\text{if } n = p^k \\text{ for a prime } p \\text{ and an integer } k \\ge 1, \\\\ 0 \u0026 \\text{otherwise}. \\end{cases}$$\nHere $\\log$ is the natural logarithm. The word prime has the meaning of the [[definition of a prime number|0SN0MAEKYX]].\n\n## Remarks\n\nFor example, $\\Lambda(8) = \\log 2$, $\\Lambda(7) = \\log 7$, and $\\Lambda(1) = \\Lambda(6) = 0$.\nFor each positive integer $n$, $\\sum_{d \\mid n} \\Lambda(d) = \\log n$.\nThe function $\\Lambda$ is not multiplicative. It is named after Hans von Mangoldt.\nThe sum $\\psi(x) = \\sum_{n \\le x} \\Lambda(n)$ is the second Chebyshev function. The [[prime number theorem|0EHBY5Y84N]] is equivalent to $\\psi(x) \\sim x$.\n\n## Explanations\n\n### L1 human\n\nThe von Mangoldt function puts the weight $\\log p$ on each prime power $p^k$, and $0$ on all other numbers.\nWith these weights, the divisor sums give exactly $\\log n$, which makes the function useful in the study of primes.\n",
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