Four Perspectives on Secondary Terms in the Davenport-Heilbronn Theorems

TL;DR

This paper reviews various approaches to understanding secondary terms in the Davenport-Heilbronn theorems related to cubic fields and class groups, highlighting historical context and methodological diversity.

Contribution

It provides a comprehensive overview of different methods used to analyze secondary terms in the Davenport-Heilbronn theorems, including historical development and recent advances.

Findings

Multiple independent proofs of secondary terms

Diverse mathematical techniques applied

Enhanced understanding of secondary term behavior

Abstract

This paper is an expanded version of the author's lecture at the Integers Conference 2011. We discuss the secondary terms in the Davenport-Heilbronn theorems on cubic fields and 3-torsion in class groups of quadratic fields. Such secondary terms had been conjectured by Datskovsky-Wright and Roberts, and proofs of these or closely related secondary terms were obtained independently by Bhargava, Shankar, and Tsimerman; Hough; Zhao; and Taniguchi and the author. In this paper we discuss the history of the problem and highlight the diverse methods used to address it.