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 $\scriptstyle a(n),\,n\,\in \,\mathbb {N} ^{+},\,$ such that

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

where $\scriptstyle m\,\perp \,n\,$ means $\scriptstyle m\,$ is coprime to $\scriptstyle n\,$. Obviously, $\scriptstyle a(1)$ 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 $\scriptstyle \alpha \geq 0$ such that $\scriptstyle a(n)=\alpha \log n$ for $\scriptstyle n\geq 1.$ 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