Let with
Prove that:
[Hint: It suffices to prove the equation above for powers of prime numbers (Why?)]
I don't know enough number theory to help you on this one.
What I would recommend you do is collect as many identities as possible and put them all in the one place: this will help you see the connections a lot quicker and should help you solve your problem quicker than say a less structured approach.
If you have all the identities and theorems on a few pages the connections should pop out.