  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.
