Power-saving error terms for the number of \(D_4\)-quartic extensions over a number field ordered by discriminant (Q6619055)

The authors show that the number of non-isomorphic quartic extensions \(K/k\) with normal closure Galois group \(D_4\) and \(\Delta_K(x)\le x\) equals \(c(k)x+O(x^{3/4})\), where\N\[\Nc(k) =\sum_{[L:k]=2}\frac1{2^{r_2(L)+1}}\frac1{\Delta^2_L}\frac{\kappa(L)}{\zeta_L(2)},\N\]\N\(\kappa(L)\) being the residue at \(s=1\) of \(\zeta_L(s)\).\N\NIn the case \(k=\mathbb Q\) the main term has been determined earlier by \textit{H. Cohen} et al. [Compos. Math. 133, No. 1, 65--93 (2002; Zbl 1050.11104)] with the error tem \(O\left(x^{3/4+\varepsilon}\right)\), and in the general case by \textit{J. Klüners} [Int. J. Number Theory 8, No. 3, 845--858 (2012; Zbl 1270.11116)] without a power-saving error term.\N\NThe authors point out that the method of the proof utilizes ideas from the paper by H. Cohen et al. quoted above.\N\NThe name of the second author of the item 18 in references is incomplete. It should be F. Diaz y Diaz.\N\NFor the entire collection see [Zbl 1540.11003].