I have "derived" (I am probably not the 1st one to do so) the following equation for the norm of Zeta^2:

Where each term j has prime factorization

, and

is the number of different prime factors. For

, the right-hand sum of converges absolutely.

I am trying to figure out how to show if the series diverges/converges or semi-converges for 0.5< a < 1. (Obviously, if the series did converge in that range that would prove the RH so it is a bit unlikely to do so...but I do not know how to approach the proof.)

The only reference to that series I could find is a much simpler form, in the special case of b=0 where it reduces to the known series (e.g. displayed in Wolfram, and in Hardy and Wright 5th edition):

Has anyone seen the above series and where? Thank you!