Secondary terms in the counting functions of quartic fields II

Abstract

We determine the smoothed counts of $S_4$-quartic fields with bounded discriminant, satisfying any finite specified set of local conditions, as the sum of two main terms with a power saving error term. We also prove an analogous result for quartic rings (weighted by the number of cubic resolvents), deducing as a consequence that the Shintani zeta functions associated to the prehomogeneous vector space ${\mathbb{C}}^2\otimes {\mathrm{Sym}}^2({\mathbb{C}}^3)$ have at most a simple pole at $s=5/6$.

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