I need to integrate $$ \int\limits_{[1,a]^n} \frac {\chi(\{ b \le x_1 \cdots x_n \le c \})} {( x_1 x_2 \cdots x_n)^k} \,dx_1 \cdots dx_n, $$ where $\chi(E)$ is the characteristic function of a set $E$. We have $a>1$ and $c>b>1$; furthermore $k\in\mathbb R$, but feel free to restrict it to either larger than $0$ or smaller than $0$ if that's easier; the case $k=0$ would also be of independent interest.

While this might look like homework, it's come up in a research context; I sincerely hope it is as easy as homework for someone, I've dug through various papers, but it seems tough. So I'm happy for any pointers!!

Thanks!

PS: I'm happy to give more context, but I'm not sure it'll add much.