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  "body": "\n## Definition\n\nThe Möbius function $\\mu$ is an arithmetic function. For each positive integer $n$, it is given by\n$$\\mu(n) = \\begin{cases} (-1)^{\\omega(n)} \u0026 \\text{if } n \\text{ is squarefree}, \\\\ 0 \u0026 \\text{otherwise}. \\end{cases}$$\nHere $\\omega(n)$ is the [[number of distinct prime factors|0YVVXT3A0B]] of $n$. The word squarefree has the meaning of the [[definition of a squarefree integer|004JW7A2N5]].\n\n## Remarks\n\nSo $\\mu(1) = 1$, $\\mu(p) = -1$ for each prime $p$, and $\\mu(n) = 0$ when $p^2 \\mid n$ for some prime $p$.\nFor example, $\\mu(6) = 1$, $\\mu(30) = -1$ and $\\mu(12) = 0$.\nThe function $\\mu$ is [[multiplicative|0VB9G5B034]], and it is the [[inverse of the constant function 1|0ASC73QEQT]] for the Dirichlet convolution.\n\n## Explanations\n\n### L1 human\n\nThe Möbius function is $0$ on numbers with a repeated prime factor.\nOn the other numbers, it is $1$ for an even number of prime factors and $-1$ for an odd number.\nIt is the sign rule of inclusion and exclusion, written as a function of $n$.\n",
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