This question may be an exact replicate of some earlier question elsewhere.
I want to know if there are better bounds than the Minkowski bound for the norm of the smallest ideal in an ideal class of a number field, which is effective when $n$ is small but seems not very sharp when $n$ is large. I noticed the crucial part of Minkowski's bound comes from his theorem on the lattice points in $\mathbb{R}^{n}$, since the theorem is sharp (see wikipedia) there is no room for improvement in that direction. My friends told me there is the Bach bound which nevertheless assumes GRH and seems not so effective when $n$ is small.
So I want to know if there is any particular method that could have helped to bound the class number from above and below in statistic terms (so that for $\mathcal{A}\cap \mathbb{Q}[\sqrt{n}]$ with $n>m$, the probability that the class number of the number field to be less than $f(m)$ is about $g(m)$, etc).
I did have done some search for relevant papers in the past. I ask because I hope someone can help explain the current state of research in layman's terms.