Using the Euler-Product of the $\zeta$ function and taking the $\log$ of it, it is easy to derive the equation $$\log\zeta(s) =\sum_p\sum_{k=1}^\infty\frac{p^{-ks}}{k}$$
However, what my textbook does at this point is just differentiation both sides and interchaging the differential operator with both sums on the right side, without any justification $$\frac{\zeta'}{\zeta}(s)= \frac{d}{ds}\sum_p\sum_{k=1}^\infty\frac{p^{-ks}}{k} = -\sum_p\sum_{k=1}^\infty\log (p)p^{-ks}=-\sum_{n=1}^\infty \Lambda(n)n^{-s}$$
The problem is, for interchanging differentiation and summation I need uniform convergence of the derivates on the right side, right?
Unfortunately I am not quite sure how to prove that kind of convergence. Are they obvious? The problem is that I start off with the (uniformly convergence) of the euler product and I am not sure what happens to the type of convergence in the moment I apply the logarithm. Is the resulting series still uniformly convergent? Unfortunately the author does not lose any words about the convergence... Later in the textbook the author uses absolute convergence of $\sum_{n=1}^\infty \Lambda(n)n^{-s}$, but again I do not see how this is maybe obvious, since he does not lose any words about it.
I'd really appreciate if anyone could give me some hints about how to handle the convergence problems there!