Hello Mathoverflow community, how are you doing? I just wanted to know if anything is known about the following integral:
$$K_n(m) = \overbrace{\int_0^1 \dots \int_0^1}^{n-\mathrm{times}} \left(-\frac{\ln(1-x_1x_2\cdots x_n)}{x_1 x_2 \cdots x_n}\right)^m \mathrm{d}x_1\mathrm{d}x_2 \cdots \mathrm{d}x_n$$
I would have assumed Adamchik, Flajolet, or Zudilin might have ran by this integral in any way, but I haven't found any connections yet. So far, I found that \begin{align} K_1(m) &= m\sum_{n=0}^{m-1}\left[ m-1 \atop n\right]\zeta(m+1-n), \\ K_2(m) &= \frac{m}{2}\sum_{n=0}^{m-1}\left[ m-1 \atop n\right]\times\left[(m+3-n)\zeta(m+2-n) -\sum_{r=1}^{m-1-n}\zeta(r+1)\zeta(m+1-n-r) \\ +2\sum_{j=1}^{m}S_{j+1, \, m+1-j-n} -\zeta(j+1)\zeta(m+1-j-n)\right], \end{align} where $$ S_{p,q} = \sum_{n=1}^\infty \frac{H_n^{(p)}}{n^q},$$ $H_n^{(p)}$ is the generalized harmonic numbars, and $\displaystyle\left[ m \atop n\right]$ is the unsigned Stirling numbers of the first kind. (Remark: The notation $S_{p,q}$ is used after Adamchik.) I have attempted higher $n$ though it quickly became unwieldy, and thus was reserved for another available afternoon. However, the case $n=2$ convinces me this might have a relationship with multiple zeta functions, though I am not an expert on the topic. Your input and advice is very much appreciated!
