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  "body": "\n## Definition\n\nThe Liouville function $\\lambda$ is the arithmetic function given by $\\lambda(n) = (-1)^{\\Omega(n)}$ for each positive integer $n$.\nHere $\\Omega(n)$ is the [[number of prime factors of n counted with multiplicity|08420828EY]].\n\n## Remarks\n\nFor example, $\\lambda(1) = 1$, $\\lambda(12) = -1$ and $\\lambda(36) = 1$.\nThe function $\\lambda$ is [[completely multiplicative|0GS27TKF1V]], since $\\Omega$ is completely additive.\nThe sum $\\sum_{d \\mid n} \\lambda(d)$ is $1$ if $n$ is a perfect square, and $0$ otherwise.\nOn squarefree integers, $\\lambda$ agrees with the [[Möbius function|0C8QENHYZW]].\n\n## Explanations\n\n### L1 human\n\nThe Liouville function is $1$ when $n$ has an even number of prime factors, counted with repeats, and $-1$ when the number is odd.\nUnlike the Möbius function, it is never $0$.\n",
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