Erdős–Delange theorem

Distribution of primes From Wikipedia, the free encyclopedia

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 denote the number of prime factors of an integer , counted with multiplicity, and be any irrational number. The theorem states that the real numbers are asymptotically uniformly distributed modulo 1.[1] It implies the prime number theorem.[2]

The theorem was stated without proof in 1946 by Paul Erdős, with a remark that "the proof is not easy".[3] 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.[1]

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