Completely multiplicative functions

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In number theory, completely multiplicative arithmetic functions are arithmetic functions $\scriptstyle a(n),\,n\,\in \,\mathbb {N} ^{+},\,$ such that

$a(mn)=a(m)a(n),\quad m,\,n\in \mathbb {N} ^{+}.\,$

This is to say that it doesn't matter whether $\scriptstyle m\,$ and $\scriptstyle n\,$ are coprime or not (i.e. share or don't share prime factors), as opposed to multiplicative arithmetic functions which require coprimality.

See also

• Multiplicative arithmetic functions
• Completely multiplicative arithmetic functions
• Additive arithmetic functions
• Completely additive arithmetic functions