In number theory, the Erdős–Delange theorem, named after Paul Erdős and Hubert Delange, is a theorem concerning the distribution of prime numbers.
Let denote the number of prime factors of an integer , counted with multiplicity, and be any irrational number. The theorem states that the fractional parts of the numbers are asymptotically uniformly distributed. 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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