The series for $\zeta'(s)$ about $s=1$ is:
$-(\frac1{(s-1)^2}) -\gamma_1 +\gamma_2(s-1)-\frac12\gamma_3(s-1)^2$....
where $\gamma$ are the Stieltjes constants.
I'm using this series to determine the asymptotic behavior of:
$\sum_{k=1}^{\infty}k^{1/{(k^{1+\frac1{\sqrt{x}}})}} -1$
Using the series I already have, I can show that the dominant term is x minus the first Stieltjes constant $\gamma_1.$
If I had a similar series for higher derivatives, I could determine if the asymptotic behavior, mainly prove or disprove whether the output exceeds $x+1$.
Does anyone happen to know the higher order derivative series for the Zeta function close to 1 like above, or a resource?