Heuristics for the asymptotics of the number of Sn$S_n$‐number fields

Abstract

AbstractWe give a heuristic argument supporting conjectures of Bhargava on the asymptotics of the number of ‐number fields having bounded discriminant. We then make our arguments rigorous in the case giving a new elementary proof of the Davenport–Heilbronn theorem. Our basic method is to count elements of small height in ‐fields while carefully keeping track of the index of the monogenic ring that they generate.