An improved error term for counting $D_4$-quartic fields

We prove that the number of quartic fields $K$ with discriminant $\mathrm{\mid }{\mathrm{\Delta }}_K\mathrm{\mid }\le X$ whose Galois closure is $D_4$ equals $CX+O(X^{5\mathrm{/}8+\epsilon })$, 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.