The Erdős Theorem and the Halberstam Theorem in function fields

Abstract

Introduction. For n ∈ N, define ω(n) to be the number of distinct prime divisors of n. The Tur´an Theorem [9] concerns the second moment of ω(n) and it implies a result of Hardy and Ramanujan [4] that the normal order of ω(n) is log log n. Further development of probabilistic ideas led Erd˝os and Kac [2] to prove a remarkable refinement of the Hardy Ramanujan Theorem, namely, the existence of a normal distribution for ω(n).