Additive arithmetic functions

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Not to be confused with additive functions, a term with a different meaning used in algebra.

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

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

where $m\perp n$ means $m$ is coprime to $n$. Obviously, $a(1)$ must be 0. An example is $\omega (n)$, the number of distinct prime factors of $n$.

Erdős proved that if a function is additive and increasing then there is some $\alpha \ge 0$ such that $a(n)=\alpha \log n$ for $n\ge 1.$ In particular, if the function is integer-valued then it is uniformly zero.

See also

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