Improved error estimates for the Davenport–Heilbronn theorems

We improve the error terms in the Davenport–Heilbronn theorems on counting cubic fields to O(X^2/3+ϵ). This improves on separate and independent results of the the authors and Shankar and Tsimerman. The present paper uses the analytic theory of Shintani zeta functions, and streamlines and simplifies the proof relative to previous work of Taniguchi and Thorne. We also give a second proof that uses a “discriminant-reducing identity” of Bhargava, Shankar, and Tsimerman and translates it into the language of zeta functions. We also 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 upon earlier work. As we explain, these results lead to quantitative improvements in various arithmetic applications.