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  "body": "\n## Statement\n\nLet $f$ and $g$ be [[multiplicative functions|0AP80TT660]].\nThen their [[Dirichlet convolution|0KH1N66QWW]] $f * g$ is a multiplicative function.\n\n## Remarks\n\nLet $m$ and $n$ be coprime. The proof uses that each divisor of $mn$ is a unique product of a divisor of $m$ and a divisor of $n$.\nWith $f = \\mathrm{id}_k \\colon n \\mapsto n^k$ and $g = \\mathbf{1}$, the theorem gives the [[multiplicativity of the divisor function|04WC36XKV6]], since $\\sigma_k = \\mathrm{id}_k * \\mathbf{1}$.\nThe Dirichlet inverse of a multiplicative function is also multiplicative.\n\n## Explanations\n\n### L1 human\n\nMultiplicative functions stay multiplicative when you convolve them.\nSo a new function built from known ones by divisor sums keeps the property, and you can compute it from prime powers.\n",
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