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

Abstract

Let $k\ge 1$ 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 $k\ge 1$. Moreover, we prove that the function ${\omega }_1(f)$ has normal order $\log (\text{deg}(f))$ and also satisfies the Erdős-Kac Theorem. Finally, we prove that the functions ${\omega }_k(f)$ with $k\ge 2$ do not have normal order.