In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the 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 1120 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function).
The first such distribution found is , where is the primecounting function and 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 . 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 2600 is prime (Failed to parse (syntax error): {\displaystyle \ log(10^{1000})} ≈ 2599.E035E8169131), whereas among positive integers of at most 2000 digits, about one in 5000 is prime (Failed to parse (syntax error): {\displaystyle \ log(10^{2000})} ≈ 4E77.X06EE4316262). In other words, the average gap between consecutive prime numbers among the first Template:Mvar integers is roughly Template:Math.^{[1]}
Statement[]
Let be the primecounting function that gives the number of primes less than or equal to x, for any real number x. For example, because there are five prime numbers (2, 3, 5, 7 and E) less than or equal to 10. The prime number theorem then states that is a good approximation to , in the sense that the limit of the quotient of the two functions and as x increases without bound is 1:
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 increases without bound. Instead, the theorem states that approximates in the sense that the relative error of this approximation approaches 0 as Template:Mvar increases without bound.
The prime number theorem is equivalent to the statement that the nth prime number p_{n} satisfies
the asymptotic notation meaning, again, that the relative error of this approximation approaches 0 as n increases without bound. For example, the 10^{10}th prime number is 28,314,567,73E,4X1, and rounds to 25,99E,035,E81,691, a relative error of about E.2373%.
The prime number theorem is also equivalent to
where Template:Mvar and Template:Mvar are the first and the second Chebyshev functions respectively.
Table of , , and []
~
, where
is the primecounting function and
is the natural logarithm of Template:Mvar
The last column,
, is the average prime gap below
.
It has been conjectured that for all x, but this is not true (the first counterexample is around 10^{200}).



10  5  4.9E  0.21  1.0521X9  2.49724X 
100  2X  24.E8  5.04  1.20EX59  4.29X708 
1000  1X5  174  31  1.1XE705  6.510390 
10,000  1426  125X  188  1.151531  8.X7412X 
100,000  10,852  E70E  1143  1.119E0E  E.405188 
1,000,000  X4,E20  97,X97  9,045  1.0E2785  11.9E9198 
10,000,000  89X,03E  834,141  65,XEX  1.094E36  14.3E1502 
100,000,000  7,799,88E  7,2E1,206  4X8,685  1.081321  16.9X0X08 
1,000,000,000  69,0E6,823  65,324,599  3,992,246  1.071360  19.38E617 
10,000,000,000  607,19X,347  596,58X,047  30,810,300  1.063E38  1E.977648 
100,000,000,000  5,587,E58,533  5,327,491,530  260,687,003  1.058536  22.36173X 
1,000,000,000,000  50,057,11X,90E  49,E49,443,990  2,109,896,E3E  1.05235X  24.94626E 
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