So recently I came across this function (it was in the context of Bose-Einstein condensation in Statistical Mechanics):
$F_\nu (\xi) = \frac{1}{\Gamma(\nu)}\int_0^\infty{\frac{x^{\nu-1}}{e^x/\xi -1}dx}$
Where $\Gamma$ is the Gamma function. It seems to resemble closely the integral form of the Riemann Zeta function if not for the $1/\xi$ (inverse fugacity $e^{\beta\mu}\in(0,1]$) in the integral. I have been looking for ages for it but I cannot seem to find any trace so I thought I'd ask about it here
Furthermore I have in my notes that
$F_\nu(\xi)=\sum_k{\frac{\xi^k}{k^\nu}}$ for $\nu=5/2$
Is this the case for all $\nu$ where the standard Riemann Zeta function would be defined as a generalised harmonic series? Or is it some kind of an approximation for this particular value of $\nu$?