Completely additive functions Other then the logarithm, does anybody know of any other important completely additive functions, by completely additive I mean $f(ab)=f(a)+f(b)$, for any integers $a,b$ ?

For any prime $p$, there's the $p$-adic order (or $p$-adic valuation) of $n$, written $\nu_p(n)$, defined by $\nu_p(n)=k$ if $p^k$ divides $n$ and $p^{k+1}$ doesn't. $\nu_p(mn)=\nu_p(m)+\nu_p(n)$
