Good morning! I'm writing about the fractional derivatives of $\zeta(s)$ right now and got a question. When computing the $k$-th derivative I arrive at
$$\zeta^{(k)}(s)=\mathrm{e}^{i \pi k} \sum_{n\ge2} \frac{\log ^{k} n}{n^{s}}$$
Now I would like to know one thing. When taking derivatives it's always about taking a limit. How can I justify the interchange of limits when taking each derivative? Help/proofs/etc. would be highly appreciated!
The direct way is to prove it for $\zeta^{(k-1)}(s)$ then $$|\frac{\zeta^{(k-1)}(s+h)-\zeta^{(k-1)}(s)}{h}-\sum_n (-1)^k n^{-s}\log^k n|$$ $$\le \sum_n |n^{-s} (\frac1h \int_s^{s+h}( n^{-x} \log^k n-\log^k n)dx)|\le \sum_n |n^{-s}| Ch \log^{k+1} n$$
and hence $$\lim_{h\to 0}\frac{\zeta^{(k-1)}(s+h)-\zeta^{(k-1)}(s)}{h}=\sum_n (-1)^k n^{-s}\log^k n$$