The prime number theorem addresses the question of what proportion of integers are prime numbers. It was proved at the end of the 19th century by the French mathematician Jacques Hadamard and the Belgian mathematician Charles de la Vallée-Poussin, working independently.

The theorem states that there are approximately N/ln N prime numbers between 2 and N. Here, ln N denotes the natural logarithm of N and "approximately" means that the ratio of the two quantities gets close to 1 as N gets large or, in other words, the limit of the ratio is 1 as N goes to infinity. Another, more informal, way of viewing the theorem is as saying that probability of a number of size N being prime is about 1/ln N. For example, the number of primes below 10⁹=1,000,000,000 (one billion) is 50,847,534, whereas 10⁹/ln(10⁹) is approximately 48,254,942 and the ratio is about 1.05.

The...