## 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)

### MSC:

 11R47 Other analytic theory

Zbl 0769.11037
Full Text:

### References:

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