# 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.