An error estimate for counting $S_3$-sextic number fields

TL;DR

This paper establishes a power-saving error estimate for counting $S_3$-sextic number fields, predicts a second main term, and provides numerical data supporting these findings and hinting at further lower order terms.

Contribution

It introduces a new power-saving error term for counting $S_3$-sextic fields and predicts an additional main term based on numerical evidence.

Findings

Proven power-saving remainder term for counting functions

Numerical data supports the predicted second main term

Evidence suggests existence of unexplained lower order terms

Abstract

In this note, we prove a power-saving remainder term for the function counting $S_3$-sextic number fields. We also give a prediction on the second main term. We also present numerical data on counting functions for $S_3$-sextic number fields. Our data indicates that our prediction is likely to be correct, and it also suggests the existence of additional lower order terms which we have not yet been able to explain.