A Carlitz module analogue of a conjecture of Erdős and Pomerance

Let $A=\mathbb{F}_q[T]$ be the ring of polynomials over the finite field $\mathbb{F}_q$ and $0 \neq a \in A$. Let $C$ be the $A$-Carlitz module. For a monic polynomial $m\in A$, let $C(A/mA)$ and $\bar{a}$ be the reductions of $C$ and $a$ modulo $mA$ respectively. Let $f_a(m)$ be the monic generator of the ideal $\{f\in A, C_f(\bar{a}) =\bar{0}\}$ on $C(A/mA)$. We denote by $\omega(f_a(m))$ the number of distinct monic irreducible factors of $f_a(m)$. If $q\neq 2$ or $q=2$ and $a\neq 1, T$, or $(1+T)$, we prove that there exists a normal distribution for the quantity

$$\frac{\omega(f_a(m))-\frac{1}{2}(\log \deg m)^2}{\frac{1}{\sqrt{3}}{(\log \deg m)^{3/2}}}.$$

This result is analogous to an open conjecture of Erdős and Pomerance concerning the distribution of the number of distinct prime divisors of the multiplicative order of $b$ modulo $n$, where $b$ is an integer with $\vert b\vert>1$, and $n$ a positive integer.