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' 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.

40 pages; submitted

References in corpus (1)

• Orbital L-functions for the space of binary cubic forms

Cited by in corpus (21)

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• The mean values of cubic L-functions over function fields
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• Uniform bounds for lattice point counting and partial sums of zeta functions
• Orbit Parametrizations for K3 Surfaces
• The shape of cubic fields
• Malle's Conjecture for $G\times A$, with $G=S_3,S_4,S_5$
• Equidistribution of bounded torsion CM points
• Orbital exponential sums for prehomogeneous vector spaces
• The local zeta function in enumerating quartic fields
• Binary quartic forms with bounded invariants and small Galois groups
• The value-distribution of Artin $L$-functions associated with cubic fields in conductor aspect
• On Malle's conjecture for nilpotent groups, I
• An error estimate for counting $S_3$-sextic number fields
• Eisenstein series twisted Shintani zeta function
• Composed varieties and reflection theorems of Ohno-Nakagawa type