An improved error term for counting \(D_4\)-quartic fields

We prove that the number of quartic fields \(K\) with discriminant \(|\Delta_K|\leq X\) whose Galois closure is \(D_4\) equals \(CX+O(X^{5/8+\varepsilon})\), improving the error term in a well-known result of Cohen, Diaz y Diaz, and Olivier. We prove an analogous result for counting quartic dihedral extensions over an arbitrary base field.