Secondary terms in counting functions for cubic fields

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’s 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.