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 $s=5/6$.
Categories: math.NT