Well, notice that since you know :
We then have:
But here: the generating function of
Now multiplying this by we get the Dirichlet-Series-Generating-Function of which is what you want.
I would like to prove that
is Liouville's function, defined as
Using multiplication of Dirichlet series, I think I can show that
but I cant see how (or if) this helps.
Any help or advice would be appreciated.