I'm trying to prove that if s is real and s>1 then

$\displaystyle |\log\zeta\left(s\right)-\underset{p}{\sum}p^{-s}|<\frac{1}{2}\left(1-2^{-s}\right)^{-1}\zeta\left(2s\right)$

By letting $\displaystyle s\rightarrow1$ I can then show that $\displaystyle \underset{p}{\sum}p^{-1}$ diverges (I hope).

I've tried all sorts of expressions for the left hand side of the inequality and don't seem to be able to get anywhere.

Any ideas?