Trinary Dirichlet convolution: $\sum_{abc=n} f(a)g(b) h(c)$ does not lead to anything new?

Question

Defining $*(f,g,h)(n) = \sum_{abc=n} f(a)g(b)h(c)$ for arithmetic functions $f, g, h$. We have for instance:

$*(f,g,h)(3) = $ " $(1,1,3) + (1,3,1) + (3,1,1)$ " where the tripple means the obvious substitution. Being only $3$ terms how could it be arrived at by say:

with $*$ below the regular Dirichlet binary product:

$$ g*h(3) = g(1)h(3) + g(3)h(1) \\ g*h(1) = g(1)h(1) $$

Then $$ f*(g*h)(3) = f(1)(g*h(3)) + f(3)(g*h(1)) $$

So they are equal on primes. Does this mean that they're equal on all numbers $n$?

Answer 1

yup. just $f*(g*h)$ is the same thing. Try expanding the sigma notation.