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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 ${\displaystyle \omega (n)}$ denote the number of prime factors of an integer ${\displaystyle n}$, counted with multiplicity, and ${\displaystyle \lambda }$ be any irrational number. The theorem states that the real numbers ${\displaystyle \lambda \omega (n)}$ 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.

References

1. ^ Delange, Hubert (1958), "On some arithmetical functions", Illinois Journal of Mathematics, 2: 81–87, MR 0095809
2. Bergelson, Vitaly; Richter, Florian K. (2022), "Dynamical generalizations of the prime number theorem and disjointness of additive and multiplicative semigroup actions", Duke Mathematical Journal, 171 (15): 3133–3200, arXiv:2002.03498, doi:10.1215/00127094-2022-0055, MR 4497225
3. Erdős, P. (1946), "On the distribution function of additive functions" (PDF), Annals of Mathematics, Second Series, 47: 1–20, doi:10.2307/1969031, JSTOR 1969031, MR 0015424; see remark at top of p. 2.

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