What is the general method for proving that an arithmetic function is multiplicative? I will list the definitions that I’m working with for multiplicity
An arithmetic function $f$ is said to be multiplicative if $f(mn) = f(m) f(n)$ whenever $m$ and $n$ are relatively prime positive integers. An arithmetic function $f$ is said to be completely multiplicative if $f(mn) = f(m) f(n)$ for all positive integers $m$ and $n$.
Now consider the following example from my textbook that I do not understand.
Let $g$ be an arithmetic function such that $g(1) = 1$ and $g(n) = 2^m$ where $m$ represents the number of unique prime numbers in the prime factorization of n. Now how would you show that $g$ is multiplicative, but not completely multiplicative?
The first thought I had was to use induction, but I started to get confused at the inductive step. But is this the way these type of problems must be solved?