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# Completely multiplicative functions

(Redirected from Completely multiplicative arithmetic functions)

In number theory, completely multiplicative arithmetic functions are arithmetic functions ${\displaystyle \scriptstyle a(n),\,n\,\in \,\mathbb {N} ^{+},\,}$ such that[1]
${\displaystyle a(mn)=a(m)a(n),\quad m,\,n\in \mathbb {N} ^{+}.\,}$
This is to say that it doesn't matter whether ${\displaystyle \scriptstyle m\,}$ and ${\displaystyle \scriptstyle n\,}$ are coprime or not (i.e. share or don't share prime factors), as opposed to multiplicative arithmetic functions which require coprimality.