What is the name of the function $1/\Gamma(z) \int_0^\infty \frac{x^{z -1}}{e^w-1}dw$? I recently saw



on the internet, and was curious what the name of the "pineapple" function was?
$$1/\Gamma(z) \int_0^\infty \frac{\color{red}{x}^{z-1}}{e^w-1}dw$$
I recognize the "mango" function as $\Gamma(z)$ the gamma function.  Also, does anyone know what theorem this refers to?
 A: One representation of the Riemann zeta-function
$$\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s}, \quad \operatorname{Re}s>1.$$
This theorem refers to the Riemann Hypothesis, one of the greatest unsolved problem in mathematics. The RH states:

All non-trivial zeros of the Riemann Zeta-Function lie on the critical line.

A: It possible rewrite Gamma function such as
$$\Gamma(s)= \int_{0}^{\infty} e^{-x} x^{s-1} dx = \int_{0}^{\infty} e^{-nu} (nu)^{s-1}ndu = n^s \int_{0}^{\infty} e^{-nu} u^{s-1} du $$
Then
\begin{align*}
\Gamma(s)\zeta(s) = \sum_{n=0}^{\infty} \frac{\Gamma(s)}{n^s} = \sum_{n=0}^{\infty} \int_{0}^{\infty}e^{-nx}x^{s-1} dx &= \int_{0}^{\infty} x^{s-1} \left (\sum_{n=0}^{\infty} e^{-nx} \right)  dx \\
& = \int_{0}^{\infty} \frac{x^{s-1}}{e^x-1} dx
\end{align*}
Hence, that is equivalent to Riemann Hypothesis.
A: It's a special case of the Bose-Einstein integral $F_-$. In (statistical) physics, we define the Bose-Einstein integral ($F_-$) and Fermi-Dirac integral ($F_+$) as follows:
$$F_{\mp}(s, \alpha)=\frac{1}{\Gamma(s)} \int_0^{+\infty}\frac{x^{s-1}}{e^{x+\alpha}\mp 1}\, \mathrm{d}x$$
So your function is just $z \mapsto F_-(z, 0)$.
