TITLE: Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$
QUESTION [24 upvotes]: I know that number fields have been the object of many statistical experiments.
Is there some kind of heuristics for the following?
Fix a degree $d$ and fix a bound $N$ on the coefficients of a monic integral degree $d$ polynomial $P$ (a more natural choice would probably be to bound the discriminant of $P$). Among those such polynomials which are irreducible, what is the proportion of those for which $\mathbf  Z[X]/(P)$ is the ring of integers of $\mathbf  Q[X]/(P)$? 
I wrote a very basic and naive script in pari/gp to test this and it seems to me that the proportion is about 60%. Is anything known or conjectured about this?
At least, is it easy to see that this proportion is non-zero?
What about the similar but different question: for which proportion of $P$ does the ring of integers have a power basis?

REPLY [7 votes]: The conjecture mentioned in the comments was proven in 2016 by Bhargava--Shankar--Wang:
The density of monic irreducible polynomials $f$ with integer coefficients such that $\mathbb Z[x]/(f(x))$ is the ring of integers in its fraction field is $\zeta(2)^{-1} = .607927\ldots\,\,\,$.
This is Theorem 1.2 of Squarefree values of polynomial discriminants I, https://arxiv.org/abs/1611.09806. An updated version is at http://www.math.uwaterloo.ca/~x46wang/Papers/monicsieve4.pdf.
Note that the density they use is with respect to the height, $$H(x^n+a_1x^{n-1}+\cdots+a_n):=\max(|a_1|,|a_2|^{1/2},\ldots,|a_n|^{1/n}),$$ which is a little different from the one proposed in the question on this page.