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

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

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

Erdős proved[1] that if a function is additive and increasing then there is some ${\displaystyle \scriptstyle \alpha \geq 0}$ such that ${\displaystyle \scriptstyle a(n)=\alpha \log n}$ for ${\displaystyle \scriptstyle n\geq 1.}$[2] In particular, if the function is integer-valued then it is uniformly zero.

2. For ${\displaystyle n=1}$, it is trivial, so he meant ${\displaystyle n>1}$.