Calculating a strange algebraic limes

Question

I have a problem with calculating a strange limes:

Let $ m \in \mathbb{N}$ be fixed and $q=p^n$ (a variabel prime power) for $n \in \mathbb{N}$ and $p$ prime. We define $$c_m=|\left\lbrace f \in \mathbb{F}_q[X]; f \ \text{irreducible, monic, deg}(f)=m \right\rbrace|. $$

Now I have to calculate $\lim_{q \to \infty} \frac{c_m}{q^m}$. And motivated from this limes the question is: Which magnitude has $$c_m=|\left\lbrace f \in \mathbb{F}_q[X]; f \ \text{irreducible, monic, deg}(f)=m \right\rbrace|$$ for large $q$?

Answer 1

$$\lim_{q \to \infty}\frac{c_m}{q^m} = \frac{1}{m}$$

The reason is that $$\begin{eqnarray}\frac{c_m}{q^m} &=& \frac{1}{q^m}\frac{1}{m}\sum_{d\mid m}\mu(d) q^{m/d} \\&=& \frac{1}{m}\left(1+\sum_{1<d\mid m}\mu(d)q^{m/d-m}\right).\end{eqnarray}$$ Now for $q\to \infty$ the sum has a constant number of summands, therefore $$\begin{eqnarray}\lim_{q\to\infty}\frac{c_m}{q^m} &=& \frac{1}{m}+ \frac{1}{m} \sum_{1<d\mid m}\mu(d)\lim_{q\to\infty}q^{m/d-m} \\&=&\frac{1}{m}.\end{eqnarray}$$

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Concerning the magnitude of $c_m$ we can see from the above that for $q \to \infty$ $$c_m = \frac{1}{m}q^m-O(q^\frac{m}{2})$$.