Alternative proofs that Dirichlet products are associative?

Question

Is there alternative proof of the following fact:

Dirichlet product on arithmetic function is associative.

I'm looking for something different than that given in Dirichlet's product with number theoretic functions.

Answer 1

The demonstration of associativity given by Joriki in the linked question is the "obvious" one in that recognizing that $(f * g) * h = f * (g * h)$ really boils down to recognizing that both are $$ f*g*h(n) = \sum_{abc = n} f(a)g(b)h(c). \tag{1}$$ There are some thin differences that can be made. For instance, you might first recognize that $$ f * g(n) = \sum_{ab = n} f(a)g(b) \tag{2}$$ [in comparison to the common first definition $f * g (n) = \sum_{d \mid n} f(d) g(n/d)$]. Then $$ (f * g) * h(n) = \sum_{dc = n} \sum_{ab = d} f(a)g(b)h(c),$$ which is merely a different rewriting of $(1)$ above.

A somewhat different point of view might come from thinking of $f,g,h$ as coefficients of the Dirichlet series $F(s), G(s), H(s)$, defined by $$ F(s) = \sum_{n \geq 1} \frac{f(n)}{n^s},\quad G(s) = \sum_{n \geq 1} \frac{g(n)}{n^s}, \quad H(s) = \sum_{n \geq 1} \frac{h(n)}{n^s}.$$ Then the $n$th coefficient of the Dirichlet series $F(s)G(s)$ is exactly $f * g(n)$. Similarly, the $n$th coefficient of $F(s)G(s)H(s)$ is exactly $f*g*h(n)$. And as the order of multiplication doesn't matter for $F(s)G(s)H(s)$ (since this is regular multiplication), the order of multiplication doesn't matter for $f*g*h$. From this point of view, it's clear that $f*g*h$ is as in $(1)$ and $f*g$ is as in $(2)$. For instance $$ F(s)G(s) = \sum_{m,n \geq 1} \frac{f(m)g(n)}{(mn)^s} = \sum_{n \geq 1} \frac{\sum_{ab = n} f(a)g(b)}{n^s}$$ is seen by collecting the $n$th coefficient directly. Proceeding in this way, one can show the normal characteristics of Dirichlet convolution, even though this is not a complete change of view.

[Aside: it can be helpful to remember both in practice, however. It is very common in studying analytic number theory to pass between the Dirichlet series and the coefficients themselves. The Dirichlet series contains analytic data, and the coefficients contain arithmetic data. Their interplay can be interesting and entertaining.]