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On an additive function on the set of ideals of an arbitrary number field. (English) Zbl 0822.11066

For a given \(\alpha>0\), let \({\mathfrak B}_ \alpha ({\mathfrak a})= \mathop {{\sum}^*}_{{\mathfrak p}\mid {\mathfrak a}} N({\mathfrak p})^ \alpha\) be the function defined on the integral ideals of an arbitrary field \(K\), where * denotes that \({\mathfrak p}\) runs over the prime ideals of \(K\). For any fixed \(N\geq 1\) it is shown that \[ \sum_{N({\mathfrak a})\leq x} {\mathfrak B}_ \alpha ({\mathfrak a})= \sum_{n=1}^ N \sum_{v=0}^{n-1} {{(- 1)^ v (1+\alpha)^ v} \over {v!}} \zeta_ K^{(v)} (1+\alpha) {{(n+1)! x^{1+\alpha}} \over {(1+\alpha)^ n\log^ n x}}+ O \Biggl( {{x^{1+\alpha}} \over {\log^{N+1} x}} \Biggr), \] where \(\zeta_ K\) is the Dedekind zeta-function of \(K\). The result sharpens and generalizes the results of J.-M. De Koninck and the reviewer [Arch. Math. 43, 37-43 (1984; Zbl 0519.10027)] and P. Zarzycki [Acta Arith. 52, 75- 90 (1989; Zbl 0682.10032)].
Reviewer: A.Ivić (Beograd)

MSC:

11N37 Asymptotic results on arithmetic functions
11R44 Distribution of prime ideals
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