Prime Number Theorem

The prime number theorem describes the asymptotic distribution of prime numbers among the positive integers, stating that the number of primes less than or equal to a given number x is approximately x divided by the natural logarithm of x, with the ratio of the two approaching exactly one as x grows toward infinity. Carl Friedrich Gauss and Adrien-Marie Legendre each independently conjectured versions of the theorem near the end of the eighteenth century based on numerical observation of how the density of primes appeared to thin out among larger numbers, but neither supplied a proof. The theorem was finally proved independently in 1896 by French mathematician Jacques Hadamard and Belgian mathematician Charles Jean de la Vallee Poussin, both relying on properties of the Riemann zeta function that Bernhard Riemann had explored in an influential 1859 paper, and a significantly simpler elementary proof avoiding complex analysis was found independently by Atle Selberg and Paul Erdos in 1949. The theorem is a central result of analytic number theory, and its refinement remains closely tied to the unresolved Riemann hypothesis, since a proof of that hypothesis would yield a much more precise bound on the theorem's own margin of error.