The Erdős–Delange theorem is a theorem in

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

concerning the distribution of

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s. It is named after

Paul Erdős Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, g ...

and Hubert Delange. Let

\omega(n)

denote the number of prime factors of an integer

n

, counted with multiplicity, and

\lambda

be any

irrational number In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...

. The theorem states that the real numbers

\lambda\omega(n)

are asymptotically uniformly distributed modulo 1. It implies the

prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, 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 p ...

. The theorem was stated without proof in 1946 by

, with a remark that "the proof is not easy". Hubert Delange found a simpler proof and published it in 1958, together with two other ways of deducing it from results of Erdős and of

Atle Selberg Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded ...

References

Theorems about prime numbers Paul Erdős {{Numtheory-stub