Dirchlet Convolution is Associative

The following is one of the properties of Dirichlet convolutions. The commutative law was proven by writing  ` n ` as a product of two factors i.e. ` n = cd `, and writing ` c ` instead of the more cumbersome `\frac{n}{d}`.  We extend this technique to three factors here.  By a summation over ` n = uvw `, I mean: take all possible combinations of ` u `, ` v ` and ` w ` as long as ` n = uvw `.


Theorem

Proof:-
For any ` n ` we have
   ` [(f \mbox{*} g) \mbox{*} h](n)  `
= ` \sum_{d|n}(f \mbox{*} g)(d) \ h(n/d) `
= ` \sum_{n=dw}(f \mbox{*} g)(d) \  h(w) `
= ` \sum_{n=dw} { \sum_{d=uv} f(u) \  g(v) } \  h(w) `
= ` \sum_{n=uvw} f(u) \  g(v) \  h(w) `
= ` \sum_{n=uc} f(u)  \  { \sum_{c=vw} g(v) \  h(w) } `
= ` \sum_{n=uc} f(u)  \  (g \mbox{*} h)(c) `
= ` [f \mbox{*} (g \mbox{*} h)](n)  `
(proven)