An improved error term for counting \(D_4\)-quartic fields (Q6644168)

Write \(N_4(D_4, X)\) for the number of quartic \(D_4\)-extensions of \(\mathbb{Q}\) with discriminant bounded by \(X\). A classical result of Cohen-Diaz y Diaz-Olivier [\textit{H. Cohen} et al., Compos. Math. 133, No. 1, 65--93 (2002; Zbl 1050.11104)] establishes for every \(\varepsilon > 0\) the asymptotic formula \N\[\NN_4(D_4, X) = CX + O_\varepsilon(X^{3/4 + \varepsilon}) \N\]\Nwith an explicit constant \(C > 0\). Cohen-Diaz y Diaz-Olivier later indicated, based on some numerical computations, that an error term of shape \(O_\varepsilon(X^{1/2 + \varepsilon})\) may be true.\N\NThe authors in this paper make significant progress by establishing an error term of the shape \(O_\varepsilon(X^{5/8 + \varepsilon})\) in their Theorem 1. Moreover, they also obtain strong results when \(\mathbb{Q}\) is replaced by a general number field \(K\). One pleasant feature for future applications is that they make the dependence on \(K\) in the error terms fully explicit, see Theorem 3.\N\NThe main new input is an explicit Polya-Vinogradov type estimate for ray class group characters. These estimates are proven in Section 4 of the paper.