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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 a random number nearby some large number N is selected, 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. In other words, the average gap between prime numbers near N is roughly ln(N).^{[1]}
Contents
Statement of the theorem
Let π(x) be the primecounting function that gives the number of primes less than or equal to x, for any real number x. For example, π(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that the limit of the quotient of the two functions π(x) and x / ln(x) as x approaches infinity is 1, which is expressed by the formula
known as the asymptotic law of distribution of prime numbers. Using asymptotic notation this result can be restated as
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