On the number of irreducible factors with a given multiplicity in function fields

Abstract

Let k ≥ 1 be a natural number and f ∈ Fq[t] be a monic polynomial. Let ωk(f) denote the number of distinct monic irreducible factors of f with multiplicity k. We obtain asymptotic estimates for the first and the second moments of ωk(f) with k ≥ 1. Moreover, we prove that the function ω1(f) has normal order log(deg(f)) and also satisfies the Erdős-Kac Theorem. Finally, we prove that the functions ωk(f) with k ≥ 2 do not have normal order.