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

TL;DR

This paper studies the distribution of irreducible factors with specific multiplicities in polynomials over finite fields, deriving asymptotic moments, normal order results, and probabilistic theorems.

Contribution

It provides new asymptotic estimates for moments of irreducible factor counts and establishes probabilistic properties for these functions over function fields.

Findings

Asymptotic estimates for the first and second moments of _k(f)

Normal order of _1(f) is _1(f) ((f)) and it satisfies the Erd-Kac Theorem

Functions _k(f) for k  2 do not have normal order

Abstract

Let $k \geq 1$ be a natural number and $f \in \mathbb{F}_q[t]$ be a monic polynomial. Let $\omega_k(f)$ 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 $\omega_k(f)$ with $k \geq 1$. Moreover, we prove that the function $\omega_1(f)$ has normal order $\log (\text{deg}(f))$ and also satisfies the Erd\H{o}s-Kac Theorem. Finally, we prove that the functions $\omega_k(f)$ with $k \geq 2$ do not have normal order.