This question on random versions of deterministic problems reminded me that many conditional results in number theory hold if the primes are in some sense random, and it is common knowledge that the Riemann hypothesis is tied up in this line of thought.

Now, off the top of my head it's not clear what "randomly distributed" really means in the context of the primes. I know that the RH would imply better versions of the prime number theorem, but that's not quite the same. So I am interested in a clarification of this as well as the following

Main question: what results would imply that, and what results would we gain from, the assumption that the primes are "randomly distributed"?