How can you prove that the convolution product of arithmetical functions is associative and that it is distributive with respect to the addition?
The book that I am reading states that $(F_a, *)$ is a commutative monoid, where $F_a$ is the set of all arithmetic functions, that is: $f: N^* \rightarrow C,$ and * is the Dirichlet product of convolution, that is: $(f*g)(n)=$the sum of all $f(d)g(n/d)$ taken for all the divisors "$d$" of $n.$
An edit with mathematical expressions would be greatly appreciated.