In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such distribution found is π(N) ~ N/log(N), where π(N) is the prime-counting function (the number of primes less than or equal to N) and log(N) is the natural logarithm of N. This means that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log(N). In other words, the average gap between consecutive prime numbers among the first N integers is roughly log(N). Consequently, a random integer with at most 2n digits (for large enough n) is about half as likely to be prime as a random integer with at most n digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log(101000) ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (log(102000) ≈ 4605.2).
Statement
Let π(x) be the prime-counting function defined to be 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 x / log x is a good approximation to π(x) (where log here means the natural logarithm), in the sense that the limit of the quotient of the two functions π(x) and x / log x as x increases without bound is 1:
lim x → ∞ π ( x ) [ x log ( x ) ] = 1 , {\displaystyle \lim _{x\to \infty }{\frac {\;\pi (x)\;}{\;\left[{\frac {x}{\log(x)}}\right]\;}}=1,}
known as the asymptotic law of distribution of prime numbers. Using asymptotic notation this result can be restated as
π ( x ) ∼ x log x . {\displaystyle \pi (x)\sim {\frac {x}{\log x}}.}
This notation (and the theorem) does not say anything about the limit of the difference of the two functions as x increases without bound. Instead, the theorem states that x / log x approximates π(x) in the sense that the relative error of this approximation approaches 0 as x increases without bound. The prime number theorem is equivalent to the statement that the nth prime number pn satisfies
p n ∼ n log ( n ) , {\displaystyle p_{n}\sim n\log(n),}
the asymptotic notation meaning, again, that the relative error of this approximation approaches 0 as n increases without bound. For example, the 2×1017th prime number is 8512677386048191063, and (2×1017)log(2×1017) rounds to 7967418752291744388, a relative error of about 6.4%. On the other hand, the following asymptotic relations are logically equivalent:
lim x → ∞ π ( x ) log x x = 1 , and lim x → ∞ π ( x ) log π ( x ) x = 1. {\displaystyle {\begin{aligned}\lim _{x\rightarrow \infty }{\frac {\pi (x)\log x}{x}}&=1,{\text{ and}}\\\lim _{x\rightarrow \infty }{\frac {\pi (x)\log \pi (x)}{x}}\,&=1.\end{aligned}}}
As outlined below, the prime number theorem is also equivalent to
lim x → ∞ ϑ ( x ) x = lim x → ∞ ψ ( x ) x = 1 , {\displaystyle \lim _{x\to \infty }{\frac {\vartheta (x)}{x}}=\lim _{x\to \infty }{\frac {\psi (x)}{x}}=1,}
where ϑ and ψ are the first and the second Chebyshev functions respectively, and to
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