Estimation of an expression involving Riemann zeta function

Question

Let $\zeta$ be the Riemann zeta function and let $\zeta^{(k)}$ be the $k-$ th derivative of the Riemann zeta function. For $s$ on the vertical line $\Re(s)=1+\kappa,$ where $\kappa$ is a positive real number,
can we have an explicit value for $\frac{\zeta^{(k)}(1-s)}{\zeta(s)}$ or are there some estimations for the term $\frac{\zeta^{(k)}(1-s)}{\zeta(s)}$?

Many thanks.

Answer 1

Well, we have the Riemann zeta function's Laurent series:

$$\zeta(s)=\frac1{s-1}+\sum _{n=0}^\infty\frac{(-1)^n\gamma _n}{n!}(s-1)^n$$

Differentiate both sides $p$ times to get

$$\zeta^{(p)}(s)=\frac{(-1)^pp!}{(s-1)^{p+1}}+\sum _{n=p}^\infty\frac{(-1)^n\gamma _n}{(n-p)!}(s-1)^{n-p}$$

which converges expecially well for $s\approx1$.