Originally Posted by

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

$\displaystyle \| \zeta \left( a, ib \right) \| ^2=\zeta \left( 2a, 0 \right) \left( 1 + \sum^{\infty}_{j=2} \frac{2^{\omega \left(j\right) } \cos(n_1 b \ln(p_1)) \ldots \cos(n_k b \ln(p_k))}{j^a} \right) $

Where each term j has prime factorization $\displaystyle j={p_1}^{n_1} {p_2}^{n_2}...$, and $\displaystyle \omega \left(j\right)$ is the number of different prime factors. For $\displaystyle a=Re\left( s \right) > 1$, 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):

$\displaystyle \frac { \| \zeta \left( a \right) \| ^2 } { \zeta \left( 2a \right) } = \sum^{\infty}_{j=1} \frac{2^{\omega \left(j\right) }} {j^a} $

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