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 $a,b$ ?

There is a general description of such functions. For each prime p, choose $f(p)$ arbitrarily. Then for a general integer $n=\prod_ip_i^{n_i}$, we have $f(n)=\sum_i n_i f(p_i)$. Possible choices for $f(p)$ include log $p$ (which gives the logarthm) and 1 for every p (the weighted prime divisor counting function) but there are uncountably many more choices.
(PS: Gerry's example of the "$p$-adic order function" comes from setting $f(p) = 1$ for one specific prime $p$ and $f(q) = 0$ for all other primes $q$.)
