Prime Number Theorem - Statement of The Theorem

Statement of The Theorem

Let π(x) be the prime-counting 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

This notation (and the theorem) does not say anything about the limit of the difference of the two functions as x approaches infinity. (Indeed, the behavior of this difference is very complicated and related to the Riemann hypothesis.) Instead, the theorem states that x/ln(x) approximates π(x) in the sense that the relative error of this approximation approaches 0 as x approaches infinity.

The prime number theorem is equivalent to the statement that the nth prime number p_n is approximately equal to n ln(n), again with the relative error of this approximation approaching 0 as n approaches infinity.