For cubic extensions of F_q(t), the discriminant-norm count is c1 q^{2N} minus c2^{N mod 3} q^{5N/3}, plus an error O(N^4 q^{3N/2}), with all constants explicit.
Secondary terms in the counting functions of quartic fields
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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
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The geometry of secondary terms in arithmetic statistics
For cubic extensions of F_q(t), the discriminant-norm count is c1 q^{2N} minus c2^{N mod 3} q^{5N/3}, plus an error O(N^4 q^{3N/2}), with all constants explicit.