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

Abstract

Let $k \geq 1$ be a natural number and $f \in \mathbb{F}_q[t]$ be a monic polynomial. Let $ω_k(f)$ 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 $ω_k(f)$ with $k \geq 1$. Moreover, we prove that the function $ω_1(f)$ has normal order $\log (\text{deg}(f))$ and also satisfies the Erdős-Kac Theorem. Finally, we prove that the functions $ω_k(f)$ with $k \geq 2$ do not have normal order.