How to remember definition of *additive* arithmetic function

Question

An arithmetic function is simply a function $f: \mathbb{N} \to \mathbb{C}$. (We're using the convention $0 \notin \mathbb{N}$.)

An arithmetic function $f$ is called multiplicative if $\forall m,n \in \mathbb{N}[gcd(m,n)=1 \to f(mn)=f(m)f(n)]$.

This definition seems easy enough to remember, as $f$ takes products to products in a nice way, at least some of the time. The definition for an additive function, however, seems a little harder to remember.

An arithmetic function $f$ is called additive if $\forall m,n \in \mathbb{N}[gcd(m,n)=1 \to f(mn)=f(m)+f(m)].$

Is there an easy way to remember that "additive" means $f(mn)=f(m)+f(m)$? With my poor memory, how do I avoid making the stupid mistake that it means $f(m+n)=f(m)f(n)$? Or that it means $f(m+n)=f(m)+f(n)$?

Answer 1

Yes, the way to remember it is to know that logarithm is the poster child for an additive function.

Answer 2

Since you're okay with multiplicative functions, you may remember additive functions using the following trick.

If you have an additive function $f$, you can make a multiplicative function $g$ by: $$g(n)=2^{f(n)}$$

With results like above, you make more connections in your brain. When you forget a few things, but if you happen to remember a result like above, you can easily cookup the definition by seeing that when f(ab) = f(a) + f(b) the function g is multiplicative.

Answer 3

Mnemonic: f stands for "factoring". Factoring is additive, in that the exponent for each factor will be the sum of the sum of the exponents of the input. If $f_p$ is the power of a factor $p$, then $f_p(p^a*p^b)=f_p(p^a)+f_p(p^b)$.

For instance, factoring $12$ gives you $2,1,0$ as the exponents of the prime numbers $2,3,5$ respectively. Factoring $45$ gives you $0,2,1$. Factoring $12*45$ gives you $2+0,1+2,0+1$.