Additive arithmetic function

An arithmetic function $f$ is additive when $f(mn)=f(m)+f(n)$ for coprime integers $m,n$. Equivalently, $f(n)={\sum }_{p^k\parallel n}f(p^k)$. This concerns multiplication of integers, and differs from an additive function on a vector space.

For an additive arithmetic function, the model mean is $A_f(x)={\sum }_{p^k\le x}f(p^k)(1-1/p)/p^k$. The exact uniform expectation differs by $O(x^{-1}{\sum }_{p^k\le x}|f(p^k)|)$.

A strongly additive arithmetic function satisfies $f(p^k)=f(p)$ for every prime $p$ and $k\ge 1$, so $f(n)={\sum }_{p\mid n}f(p)$. The prime omega function is an example.

A completely additive arithmetic function satisfies $f(mn)=f(m)+f(n)$ for all positive $m,n$, equivalently $f(p^k)=kf(p)$. The total number of prime factors is an example.