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?