Erdős–Delange theorem

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