{
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  "body": "\n## Definition\n\nAn [[arithmetic function|0H2EPRZC0D]] $f$ is completely multiplicative if $f(1) = 1$ and $f(mn) = f(m) f(n)$ for all positive integers $m$ and $n$.\n\n## Remarks\n\nEach completely multiplicative function is [[multiplicative|0AP80TT660]]. The converse fails: [[Euler's totient function|0VV04M41AN]] has $\\varphi(4) = 2$ but $\\varphi(2)^2 = 1$.\nA completely multiplicative function is determined by its values at the primes.\nExamples are $n \\mapsto n^k$, the [[Liouville function|08RQAPXF1T]] and, for a fixed odd prime $p$, the [[Legendre symbol|01J8AMV9RS]] $n \\mapsto \\left(\\frac{n}{p}\\right)$.\n\n## Explanations\n\n### L1 human\n\nA completely multiplicative function respects every product, not only products of coprime numbers.\nSo it is enough to know its values at the primes, as for the function $n \\mapsto n^2$.\n",
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