Residue of the following variant of Dirichlet function [closed]

Question

I am working with the Piltz divisor problem where the number of ways in which a number $n$ can be written as a product of $k$ is of the form

$$D_{k}(x)=xP_{k}(\log x)+\Delta _{k}(x)$$

where $P_k$ is a polynomial of degree $k-1$ defined as $$ P_{k-1}(log x) = Res_{s=1} x^{s-1} \zeta^{k}(s) s^{-1}\\ \zeta(s) = \frac{1}{s-1} + \gamma + \sum_{k=0}^{\infty}\gamma_k*(s-1)^k $$ $\gamma_k$ is probably the $k^{th}$derivative of $-\Gamma(1)$ but I'm not sure about that.

I need help with finding this residue.

Thanks in advance!

Answer 1

Have a look at p.171 of this paper by A.F. Lavrik. You should find your answer there.