Note on the number of integral ideals in Galois extensions (Q625868)

Let \(K\) be a number field and \(a_k\) be the number of integral ideals in \(K\) with norm \(k\). Landau studied the sum \(\sum_{k \leq x} a_k\) when \(K\) has degree \(\geq 2\). Several other authors have studied this problem. The best known results for number fields \(K\) of degree \(\geq 3\) is by \textit{W. G. Nowak} [Math. Nachr. 161, 59--74 (1993; Zbl 0803.11061)]. In this paper, the authors study the sum \(\sum_{k \leq x} a_k^l\) for any integer \(l \geq 2\) and when \(K\) is Galois of degree \(n \geq 2\). Their result improves an earlier result of \textit{K. Chandrasekharan} and \textit{A. Good} [Monatsh. Math. 95, 99--109 (1983; Zbl 0498.12009)]. Furthermore, if \(K\) is abelian, the authors obtain a better bound. The authors also study the number of solutions of polynomial congruences as an application of their results.