in the paper http://arxiv.org/pdf/math-ph/0005029v2.pdf formula 23 Voros evaluate and get the following spectral theta /(semiclassical) function
$ \sum (E_{n}+\lambda )^{-s}= \frac{\Gamma (s-1/2)}{\Gamma(s)\sqrt \pi}\int_{0}^{\infty}\frac{dx}{(V(x)+\lambda)^{s-1/2})}= \zeta (s, \lambda)$
$ H=p^{2}+V(x). $
i believe he has used the identity for the Gamma function $ \Gamma (s-1/2)a^{1/2-s} = \int_{0}^{\infty} dt t^{s-1/2}e^{-at} $
However how could i compute the derivative $ \frac{ \partial }{\partial s}\zeta (s, \lambda)$ at $ s=0 $ should i ignore all the divergent termes and simply take the finite part ?? thanks.
