Strictly speaking birds don't even have a cerebral cortex. That is, cortex evolved in mammals after the split with the last common ancestor of mammals and birds. So in that sense, birds would be at the very back of the ordering, along with reptiles, fish and the other vertebrates that lack a cortex. That said, birds do have a well developed pallium that seems to play a similar role as cortex does in mammals, even though it is structurally quite different. If you let pallium stand in for cortex, crows would probably do okay.
It seems likely that avian pallium performs similar computations as cortex, despite having a different implementation. The more interesting question, in my opinion, is not who has the most neurons of any particular type, but how those cells enact computations. Does the avian pallium have representations that are similar to mammalian cortex. In other words, does it solve problems in similar ways/using similar algorithms as cortex or did its early divergence allow it to find different solutions?
To me it seems our ability to preserve and share knowledge is far more important for our intellectual pursuits than the individual brain is. E.g. it took hundreds of thousands of years for someone to think up the number 0, and then it spread like wildfire (after the authorities stopped resisting).
What I find curious is that parrots can vocalize at least as well as humans, they show some pretty solid reasoning ability, their tongue is practically an opposable thumb, and yet they don't seem to have developed complex spoken language or technology.
I wonder what's missing from bird brains that would help them with language. Actually, I just had a flash of memory about some part of the brain involved with putting ideas into linear sequences, as we do with words. Maybe that plays a part.
Corvids and cetaceans would be good for future study, but I doubt we're in for much of a surprise.
This is a study in line with interspecies differences. Intraspecies differences still require some teasing out. E.g., how does a Gauss differ from Joe Q. Public?
It seems likely that avian pallium performs similar computations as cortex, despite having a different implementation. The more interesting question, in my opinion, is not who has the most neurons of any particular type, but how those cells enact computations. Does the avian pallium have representations that are similar to mammalian cortex. In other words, does it solve problems in similar ways/using similar algorithms as cortex or did its early divergence allow it to find different solutions?