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  "body": "\n## Definition\n\nLet $f$ and $g$ be [[arithmetic functions|0H2EPRZC0D]].\nTheir Dirichlet convolution is the arithmetic function $f * g$ given by\n$$(f * g)(n) = \\sum_{d \\mid n} f(d) \\, g\\!\\left(\\frac{n}{d}\\right)$$\nfor each positive integer $n$, where $d$ runs over the [[positive divisors|0FCWA16FZE]] of $n$.\n\n## Remarks\n\nThe convolution is commutative and associative, and it distributes over addition.\nThe identity element is $\\varepsilon$, with $\\varepsilon(1) = 1$ and $\\varepsilon(n) = 0$ for $n \u003e 1$.\nAn arithmetic function $f$ has an inverse for $*$ if and only if $f(1) \\ne 0$.\nFor example, with the constant function $\\mathbf{1}$, $(\\mathbf{1} * \\mathbf{1})(n)$ is the [[number of divisors|0BKJT6WG96]] of $n$.\n\n## Explanations\n\n### L1 human\n\nThe Dirichlet convolution combines two functions by a sum over the factorizations $n = d \\cdot e$.\nIt is the natural product for functions defined by sums over divisors.\n",
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