What is v(d)? Is this some kind of special analytic number theory function?
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.