On the distribution of integer ideals in algebraic number fields. (English) Zbl 0803.11061

Suppose that \(K\) is an algebraic number field of degree \(n\geq 2\) over \(\mathbb{Q}\), \(R_ K\) the ring of algebraic integers in \(K\), and \({\mathcal C}\) a fixed class of integer ideals in \(R_ K\). Let \(A(x,{\mathcal C})\) be the number of integer ideals \(a\in {\mathcal C}\) with norm \(N(a)\leq x\). E. Landau showed that \[ A(x,{\mathcal C})= x+ O(x^{1- {2\over {n+1}}}) \] for large positive \(x\). The author improves this result for \(n\geq 3\). The exponent in the error term is now reduced to \[ 1-2/n+ 8/n(5n+2) \quad \text{for} \quad 3\leq n\leq 6 \qquad \text{and to} \qquad 1-2/n+ 3/2n^ 2 \quad \text{for} \quad n\geq 7. \] The result and the method of proof is similar to the estimation of the lattice rest of a large convex body having finite nonzero Gaussian curvature throughout. See the paper of the author and the reviewer [Acta Arith. 62, 285-295 (1992; Zbl 0769.11037)].
Reviewer: E.Krätzel (Jena)


11R47 Other analytic theory


Zbl 0769.11037
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