(Note: I’m describing here a topic where I can’t personally follow the proofs. Think of this as being a second-hand account)
The prime number theorem states that asymptotically the number of primes less than is approximately . It is fairly easy to prove that this is true up to a constant, that is, letting denote the number of primes less than , that
for some constant . So the interesting thing is the constant factor.
When you think about it, it’s rather strange that the constant factor is exactly one. Consider a similar case: counting the number of twin primes less than a given . The value is conjectured to be approximately , where
This constant can be generated as follows: We want to determine the probability that are both prime for . The probability for to be prime and for to be prime are both , so assuming they are independent the probability for both being prime is . However, these probabilities are not independent. For example, assuming and are independent would lead to a probability of of them both
being odd, whereas the actual probability is . Taking this into account, the probability estimate should be twice what it was before, up to . Similarly, it is possible to calculate that the probability that a certain prime divides neither nor is different by a factor of than it would under the indepence assumption. Multiplying all these factors out you get the constant .
You would expect something similar to happen when counting primes: That there should be some constant factor that is some product over all primes, and that by default it would be a fairly random-looking number rather than exactly 1. The naive estimate fails. Still, it seems strange that the fact that each prime number is odd, which reduces the frequency of prime by a half, and that they are all 1 or 2 mod 3, which reduces by , and so, all reduce to a constant factor of exactly 1.
The explanation is this: The frequency of the primes is self-adjusting. For instance, in the interval there are less primes than there are supposed to be, that would make any number in more likely to be prime. Thus facts like how the specific number 2 is prime in the long run make no difference in the density of prime.