Secondary terms in the counting functions of quartic fields

Abstract: We prove that the smoothed counting function of the set of quartic fields, satisfying any finite set of local conditions, can be written as a linear combination of $X,X^{5/6}\log X,X^{5/6}$, upto an error term of $O(X^{13/16+o(1)})$. For certain sets of local conditions, namely, those cutting out ``$S_4$-families'' of quartic fields, we explicitly determine the leading constants of the secondary terms. We moreover express these constants in terms of secondary mass formulas associated to families of quartic fields