Heuristics for the asymptotics of the number of $S_n$-number fields

TL;DR

This paper provides heuristic support for conjectures on the growth of $S_n$-number fields with bounded discriminant, and offers a new elementary proof for the case $n=3$ using counting methods.

Contribution

It introduces heuristic arguments for Bhargava's conjectures and rigorously proves the case $n=3$ with an elementary approach.

Findings

Heuristic arguments support conjectures on $S_n$-number fields.

Rigorous proof of the Davenport-Heilbronn theorem for $n=3$.

Counting elements of small height aids in understanding field asymptotics.

Abstract

We give a heuristic argument supporting conjectures of Bhargava on the asymptotics of the number of $S_n$-number fields having bounded discriminant. We then make our arguments rigorous in the case $n=3$ giving a new elementary proof of the Davenport-Heilbronn theorem. Our basic method is to count elements of small height in $S_n$-fields while carefully keeping track of the index of the monogenic ring that they generate.