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 rough description of how the primes are distributed.Roughly speaking, the prime number theorem states that if you randomly select a number nearby some large number N, the chance of it being prime is about 1 / ln(N), where ln(N) denotes the natural logarithm of N. For example, near N = 10,000, about one in nine numbers is prime, whereas near N = 1,000,000,000, only one in every 21 numbers is prime.
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<mathematics> The number of prime numbers less than x is about x/log(x). Here "is about" means that the ratio of the two things tends to 1 as x tends to infinity. This was first conjectured by Gauss in the early 19th century, and was proved (independently) by Hadamard and de la Vall'ee Poussin in 1896. Their proofs relied on complex analysis, but Erds and Selberg later found an "elementary" proof.
(1995-04-10)