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  "body": "\n## Statement\n\nLet $f$ and $g$ be arithmetic functions.\nThen $g(n) = \\sum_{d \\mid n} f(d)$ for all positive integers $n$ if and only if\n$$f(n) = \\sum_{d \\mid n} \\mu(d) \\, g\\!\\left(\\frac{n}{d}\\right)$$\nfor all positive integers $n$. Here $\\mu$ is the [[Möbius function|0C8QENHYZW]].\n\n## Remarks\n\nIn terms of the [[Dirichlet convolution|0KH1N66QWW]], the statement is: $g = f * \\mathbf{1}$ if and only if $f = g * \\mu$.\nThe proof uses [[$\\mu * \\mathbf{1} = \\varepsilon$|0ASC73QEQT]] and the associativity of $*$.\nFor example, the [[sum of the totient over the divisors|09JXE6XMCC]] gives $\\sum_{d \\mid n} \\varphi(d) = n$. So $\\varphi(n) = \\sum_{d \\mid n} \\mu(d) \\, n/d$.\n\n## History\n\nAugust Ferdinand Möbius introduced the formula into number theory in 1832.\n\n## Explanations\n\n### L1 human\n\nIf you know all the divisor sums of $f$, the formula recovers $f$ itself.\nIt undoes the operation \"sum over the divisors\", as subtraction undoes addition.\n",
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