It is known that:
$$\Bigg| \Big(\frac{1}{\zeta(s)}\Big)^{k)} \Bigg|= \Bigg |\sum\limits_{n=1}^{\infty} (-1)^k \frac{\mu(n)}{n^s} \log^k{n} \Bigg|$$
when $\Re(s) > 1$. Can you help find a bound or asymptotic ? No matter how sharp/trivial it is. I am specially interested when $k \to \infty$.