What is the best simulation of majority utilizing $\bmod\{2,3,\dots,p\}$ gates? It is known $AC^0[2]$ cannot get majority function.
Is there a literature on simulation of $MAJ$ function utilizing $AC^0[2,3,\dots,p]$ gates for a finite fixed set of primes $2$ to $p=O(1)$?
What is the best we found?
Although not exactly what you're looking for, this paper seems to be the most current (2019) efforts towards tight $AC^0[2]$ bounds as of yet, and provides a good summary of the difference between $AC^0$ and $AC^0[2]$ bounds.