prime number theorem: a standalone construct

Carl Friedrich Gauss, in unpublished work from around 1792 and later described in his 1849 letter to Johann Encke, and Adrien-Marie Legendre, in his Essai sur la théorie des nombres (published 1798 and expanded in 1808), independently conjectured that the count of prime numbers up to a value x, written pi(x), keeps pace with x divided by the natural logarithm of x. The prime number theorem is the proved form of that observation: pi(x) is asymptotic to x/ln(x), meaning the ratio of the two quantities approaches 1 as x grows without bound, so the primes thin out at a rate set by the logarithm even though they continue forever. It was first proved in 1896, independently and in the same year, by Jacques Hadamard and Charles-Jean de la Vallée Poussin, each drawing on the analytic behaviour of the Riemann zeta function, a function of a complex variable whose zeros govern how the primes are distributed. Later mathematicians found other routes to the same result, including an elementary proof avoiding complex analysis by Atle Selberg and Paul Erdős in 1949 and a short analytic proof by Donald Newman in 1980. As a landmark, the theorem fixes the large-scale density of the primes, the basic building blocks of the integers, converting an eighteenth-century numerical guess into an exact asymptotic law.[1][2][3][4][5]

Unit

A proved asymptotic law relating pi(x), the prime-counting function, to x divided by the natural logarithm of x

Recovery means

Not applicable

Measured by

Established by mathematical proof; the approximation is checked numerically by counting the primes up to large x and comparing pi(x) with x/ln(x) or with the logarithmic integral Li(x)