With primes I think a general question is to what extent they are "random". Clearly, they are defined by a fixed rule, so they are not actually random, but they

With primes I think a general question is to what extent they are "random". Clearly, they are defined by a fixed rule, so they are not actually random, but they have a certain density, statistics, etc. So we can try to narrow down in what sense they are like random numbers and in what sense not. Do they produce a similar "clustering" or (lack thereof) as if we would generate random numbers according to some density, etc. I see the twin primes question fitting in such a theme. Happy to learn if a number theorist has more insight.