Erdős–Delange theorem

In number theory, the Erdős–Delange theorem, named after Paul Erdős and Hubert Delange, is a theorem concerning the distribution of prime numbers.

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 fractional parts of the numbers ${\displaystyle \lambda \omega (n)}$ are asymptotically uniformly distributed.^[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]