On the Mertens-Cesàro theorem for number fields (Q2800090)

Let \(K\) be an algebraic number field, \(O\) its ring of integers, \(\Omega =\{\omega_1,\dots,\omega_n\}\) an integral basis and put NEWLINE\[NEWLINEO(x,\Omega)=\{\sum_{i=1}^n a_i\omega_i:\;|a_i|\leq x, a_i\in Z\}.NEWLINE\]NEWLINE For a subset \(T\) of \(O^m\) (\(m\geq1\)) the authors define its density \(d_\Omega(T)\) by NEWLINE\[NEWLINEd_\Omega(T)=\lim_{x\to\infty}{|O(x,\Omega)^m\cap T|\over (2x)^{mn}},NEWLINE\]NEWLINE provided this limit exists.NEWLINENEWLINEThe main result of the paper states that the density of the set of all co-prime \(m\)-tuples of integers of \(K\) exists, does not depend on \(\Omega\) and equals \(1/\zeta_K(m)\), \(\zeta_K(s)\) being the Dedekind zeta-function of \(K\). This generalizes the classical result of \textit{F. Mertens} [J. Reine Angew. Math. 77, 289--339 (1873; JFM 06.0114.02)] (rediscovered later by Cesàro [Mathesis 3, 224--225 (1883)]) in the case \(K=Q\) and \(m=2\), and the result of \textit{J. E. Nymann} [J. Number Theory 4, 469--473 (1972; Zbl 0246.10038)] for \(K=Q\) and arbitrary \(m\).