## Note on the number of integral ideals in Galois extensions.(English)Zbl 1273.11160

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 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 K. Chandrasekharan and 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.

### MSC:

 11R42 Zeta functions and $$L$$-functions of number fields 11R45 Density theorems

### Citations:

Zbl 0803.11061; Zbl 0498.12009
Full Text:

### References:

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