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1 October 2013 Secondary terms in counting functions for cubic fields
Takashi Taniguchi, Frank Thorne
Duke Math. J. 162(13): 2451-2508 (1 October 2013). DOI: 10.1215/00127094-2371752

Abstract

We prove the existence of secondary terms of order X5/6 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.

Citation

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Takashi Taniguchi. Frank Thorne. "Secondary terms in counting functions for cubic fields." Duke Math. J. 162 (13) 2451 - 2508, 1 October 2013. https://doi.org/10.1215/00127094-2371752

Information

Published: 1 October 2013
First available in Project Euclid: 8 October 2013

zbMATH: 1294.11192
MathSciNet: MR3127806
Digital Object Identifier: 10.1215/00127094-2371752

Subjects:
Primary: 11R16
Secondary: 11R29, 11R45

Rights: Copyright © 2013 Duke University Press

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Vol.162 • No. 13 • 1 October 2013
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