In [1] it is stated that
"It remains an open question as to whether every function in $\#P$ has $TC^0$ circuits (although it is at least known that not all $\#P$ functions have DLogTime-uniform $TC^0$ circuits)."
$TC^0$ circuits generated by DLogTime functions does not contain $\#P$. We don't know if $TC^0$ circuits generated by arbitrary functions does not contain $\#P$.
Is there anything known about the cases in between these two? E.g. is it known if $TC^0$ circuits generated by $L$ does not contain $\#P$?
- [1] Agarwal, Allender, and Datta, "On $TC^0$, $AC^0$, and Arithmetic Circuits"