Improved error estimates for the Davenport-Heilbronn theorems

TL;DR

This paper refines the error estimates in counting cubic fields, achieving a sharper bound of $O(X^{2/3 + psilon})$, using advanced analytic techniques and simplifying previous proofs.

Contribution

It introduces improved error bounds for the Davenport-Heilbronn theorems, employing the analytic theory of Shintani zeta functions and a novel discriminant-reducing identity.

Findings

Error term improved to $O(X^{2/3 + psilon})$

Provides explicit error dependence for local conditions

Simplifies previous zeta function proofs

Abstract

We improve the error terms in the Davenport-Heilbronn theorems on counting cubic fields to $O(X^{2/3 + \epsilon})$. This improves on separate and independent results of the authors and Shankar and Tsimerman. The present paper uses the analytic theory of Shintani zeta functions, and streamlines and simplifies the earlier zeta function proof. We also give a second proof that uses a "discriminant-reducing identity" and translates it into the language of zeta functions. We additionally provide a version of our theorem that counts cubic fields satisfying an arbitrary finite set of local conditions, or even suitable infinite sets of local conditions, where the dependence of the error term on these conditions is described explicitly and significantly improves on our previous works. As we explain, these results lead to quantitative improvements in various arithmetic applications.