I was working on a proof to determine that a product cannot be done in AC0, how can a proof that can be done in AC1?
Multiplication can be done even with stronger restrictions, like $AC^1$ with bounded fan-in.
Proof is little hard to typeset here, but I will outline the sketch and give a link.
- You shall prove, that addition of two m-bit numbers have $O(m)$ size and constant depth circuit and thus is in $AC^0$. This is pretty simple (start with looking for $O(m)$ depth and then simplify)
- Then you need to generalize this result to addition of many m-bit numbers and show that it have logarithmic depth and thus in $AC^1$
- And finally what you are looking for (product in $AC^1$) is simple corollary
Look into those lecture notes for example, where this proof is done in great details.