The Erdős–Delange theorem is a theorem in number theory concerning the distribution of prime numbers. It is named after Paul Erdős and Hubert Delange. Let ω ( n ) {\displaystyle \omega (n)} denote the number of prime factors of an integer n {\displaystyle n} , counted with multiplicity, and λ {\displaystyle \lambda } be any irrational number. The theorem states that the real numbers λ ω ( n ) {\displaystyle \lambda \omega (n)} are asymptotically uniformly distributed modulo 1. It implies the prime number theorem. The theorem was stated without proof in 1946 by Paul Erdős, 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.
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