I have just proved that
$$- \frac{\zeta'(s)}{\zeta(s)} = \sum\limits_p\frac{\log(p)}{p^s - 1}$$ and am aiming to prove that
$$ -\frac{\zeta'(s)}{\zeta(s)} = \sum\limits_p\frac{\log(p)}{p^s} + h(s) $$
with $ h $ being a holomorphic function on $ \Re(s) > 0 $.
I have calculated that $ h(s) = \sum\limits_p \frac{\log(p)}{p^s(p^s-1)} $. I just don't know how to prove that it is holomorphic there - I'm not very familiar with complex analysis and I crossed this mooching around "analytic number theory".
I'd appreciate some help