Secondary terms in counting functions for cubic fields

TL;DR

This paper establishes the existence of secondary terms in counting functions for cubic fields, confirming conjectures and revealing biases, using zeta function analysis related to binary cubic forms.

Contribution

It proves secondary terms in cubic field counting functions, confirming conjectures and extending results to arithmetic progressions with bias analysis, using zeta function techniques.

Findings

Confirmation of secondary terms of order X^{5/6} in cubic field counts

Discovery of bias in secondary terms in arithmetic progressions

Independent proof of Roberts' conjecture by other researchers

Abstract

We prove the existence of secondary terms of order X^{5/6} in the Davenport-Heilbronn theorems on cubic fields and 3-torsion in class groups of quadratic fields. For cubic fields this confirms a conjecture of Datskovsky-Wright and Roberts. We also prove a variety of generalizations, including to arithmetic progressions, where we discover a curious bias in the secondary term. Roberts' conjecture has also been proved independently by Bhargava, Shankar, and Tsimerman. In contrast to their work, our proof uses the analytic theory of zeta functions associated to the space of binary cubic forms, developed by Shintani and Datskovsky-Wright.