I am working with the [Piltz divisor problem](https://en.wikipedia.org/wiki/Divisor_summatory_function) 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](https://arxiv.org/pdf/1711.09589.pdf) 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!



Residue of the following variant of Dirichlet function
Have a look at p.171 of this [paper](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tm&paperid=2566&option_lang=eng) by A.F. Lavrik. You should find your answer there.


