TITLE: Probability of $\mathcal{O}_{\mathbb{Q}(\alpha)}=\mathbb{Z}[\alpha]$
QUESTION [11 upvotes]: Let $f(x)\in \mathbb{Z}[x]$ be a irreducible polynomial of degree $n$ and let $\alpha$ be a root of $f(x)$. One can show that $\mathcal{O}_{\mathbb{Q}(\alpha)}=\mathbb{Z}[\alpha]$ if discriminant of $f(x)$ is square-free, and converse does not hold in general. I want to know the probability of this event, i.e. 
$$
p_{n}=\lim_{N\to\infty} \frac{\#\{f(x):\mathcal{O}_{\mathbb{Q}(\alpha)}=\mathbb{Z}[\alpha],\alpha\text{ is a zero of $f(x)$}\}}{\#\{f(x)\in \mathbb{Z}[x]:f(x)\text{ is monic irreducible polynomial of degree $n$}, h(f)\leq N\}}
$$
where $h(f)=\max\{|a_{0}|, \dots, |a_{n-1}|\}$ for $f(x)=x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}$. I don't know whether the limit exists or not, but as I explained above, lower bound of the limit is 
$$
q_{n}=\lim_{N\to\infty} \frac{\#\{f(x):disc(f)\text{ is square-free}\}}{\#\{f(x)\in \mathbb{Z}[x]:f(x)\text{ is monic irreducible polynomial of degree $n$}, h(f)\leq N\}}.
$$
For example, in case of $n=2$, we have
$$
q_{2} = \lim_{N\to\infty}\frac{\{a, b\in \mathbb{Z}:|a|, |b|\leq N, a^{2}-4b\text{ is square-free}\}}{\{a, b\in \mathbb{Z}:|a|, |b|\leq N, a^{2}-4b\text{ is not square}\}}
$$

Edit : With computer, I computed $q_{2}$ and 
N=10 -> 0.6
N=100 -> 0.454497
N=500 -> 0.425595
N=1000 -> 0.419174
N=5000 -> 0.41124

and my computer doesn't work for $N=10000$..

REPLY [7 votes]: You are in luck. This exact problem has been considered in this paper: https://arxiv.org/abs/1611.09806 . They prove that $ p_n = 1/\zeta(2)$, in particular it is independent of $n$! 
Read the introduction to the paper, it is very clear. 
(Do note that they work with a different height function.)