Prime Number Theorem

In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers.

Informally speaking, the prime number theorem states that if a random integer is selected in the range of zero to some large integer N, the probability that the selected integer is prime is about 1 / ln(N), where ln(N) is the natural logarithm of N. For example, among the positive integers up to and including N = 103 about one in seven numbers is prime, whereas up to and including N = 1010 about one in 23 numbers is prime (where ln(103)= 6.90775528. and ln(1010)=23.0258509). In other words, the average gap between consecutive prime numbers among the first N integers is roughly ln(N).