Completely additive arithmetic functions

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

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

Obviously, $\scriptstyle a(1)$ must be 0. An example is $\Omega (n)$, the number of prime factors of $n$.

See also

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