On the number of irreducible factors with a given multiplicity in function fields - Das, Sourabhashis; Elma, Ertan; Kuo, Wentang; Liu, Yu-Ru (Elsevier, 2023)
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.
The final publication is available at Elsevier via https://doi.org/10.1016/j.ffa.2023.102281. © 2023. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Keywords: monic irreducible factors; normal order; Erdős-Kac theorem