Multidimensional small divisor functions

This is a short note generalizing the construction from arXiv:1906.07410, arXiv:2009.04955 to multi-indices. We recommend to consider both references first. We obtain polar harmonic Maa{\ss} forms of non-positive integral weight if the dimension is even and greater than $2$. We provide explicit examples in dimension $4$, $6$, $8$, and $10$.