I looked up arithmetic functions, originally to see if there is a chart that clearly listed the multiplicative functions we've been talking about (because I tend to get them all mixed up in my head). But, I ended up coming to the wikipedia page about arithmetic functions, and I noticed that, in addition to multiplicative and completely multiplicative properties of arithmetic functions, arithmetic functions can also be additive and completely additive. Here's what Wikipedia told me:

Let (m,n) denote the greatest common divisor of m and n. An arithmetic function a is said to be

* multiplicative if a(mn) = a(m)a(n) for all natural numbers m and n such that (m,n) = 1;

* completely multiplicative if a(mn) = a(m)a(n) for all natural numbers m and n;

* additive if a(mn) = a(m)+a(n) for all natural numbers m and n such that (m,n) = 1;

* completely additive if a(mn) = a(m)+a(n) for all natural numbers m and n.

The definition of additive makes a lot of sense compared to the definition of multiplicative. So then I started wondering about additive functions and why we haven't talked about them. So I followed a link to the additive functions page on wikipedia. As it turns out, a couple of the multiplicative functions are also additive… however, I couldn't come up with any good uses of additive functions, or anything particularly interesting about them from the wikipedia page. Can anybody else find anything?