# analytic continuation of an integral involving the mittag-leffler function

i have posted this question on MO, and i didn't get an answer . we have the following integral :

$$I(s)=\int_{0}^{\infty} \frac{s}{2x}\left(E_{s/2}((\pi x)^{s/2})-1\right)\omega(x)dx -\lim_{z \to 1 }\zeta(z)$$

where : $E_{\alpha}(z)$ is the mittag-leffler fuction

$\omega(x)=\frac{\theta(ix)-1}{2}$

$\theta(x)$ is the jacobi theta function

and $\zeta(s)$ is the Riemann zeta function

the integral behaves well for $Re(s)>1$ . i am trying to extend the domain of the integral to the whole complex plane except for some points. i have tried the identities:

$\omega(x^{-1})=-\frac{1}{2}+\frac{1}{2}x^{1/2}+x^{1/2}\omega(x)$

and

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

to split the integration at 1 , and replace $x$ by $x^{-1}$ in the interval [0,1], and hoped to get a factor that would cancel out with the divergent $\zeta(z)$, and get the Euler–Mascheroni constant or something . but it seems i am lost !!! hence, the post .

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