greetings . we have the integral :

$I(s)=\int_{0}^{\infty}\frac{s(E_{s}(x^{s})-1)-x}{x(e^{x}-1)}dx $

$E_{\alpha}(z)$ being the mittag-leffler function, which is defined for Re(s)>0

the integral above behaves well for Re(s)>1 . i am trying to extend the domain of $I(s)$ to the whole complex plane except for some points. but i have no idea where to start !!

the mittag-leffler function admits the beautiful continuation :

$E_{\alpha}(z)=1-E_{-\alpha}(z^{-1}) $

using the fact that $I(s)=\frac{1}{4}\int_{0}^{\infty}\frac{\theta(ix) \left(sE_{s/2} ((\pi x)^{s/2})-s-2x^{1/2}\right)}{x}dx $

and :

$\theta(-\frac{1}{t})=(-it)^{1/2}\theta(t) $

one can split the integration much like the one concerning the Riemann zeta . but i am not sure this will yield a meromorphic integral . hence, the problem !!