×

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

Citations:

Zbl 0769.11037
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chandrasekharan, Math. Ann. 152 pp 30– (1963)
[2] Hafner, J. Number Th. 17 pp 183– (1983)
[3] Huxley, Proc. London Math. Soc. (3) 60 pp 471– (1990)
[4] Huxley, Proc. London Math. Soc.
[5] : The Riemann zeta-function. New York 1985
[6] Ivić, Acta Arithm. 52 pp 241– (1989)
[7] : On the estimation of multiple exponential sums. In: Recent progress in analytic number theory, vol I, pp. 231–246 (eds. H. Halberstam and C. Hooley). London 1981
[8] : Lattice points. Dordrecht–Boston–London 1988 · Zbl 0675.10031
[9] Krätzel, Acta arithm. 62 pp 285– (1992)
[10] : Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale. 2nd ed. New York 1949 · Zbl 0045.32202
[11] : Geometry of numbers. Amsterdam–New York–Oxford–Tokyo 1969
[12] Müller, Monatsh. f. Math. 106 pp 211– (1988)
[13] : Elementary and analytic theory of algebraic numbers. Warszaw 1974 · Zbl 0276.12002
[14] Titchmarsh, Proc. London Math. Soc (2) 36 pp 485– (1934)
[15] Titchmarsh, London Math. Soc. (2) 38 pp 96– (1934)
[16] : Lehrbuch der Algebra, vol II. Braunschweig 1896
[17] : Zetafunktionen und quadratische Körper. Berlin–Heidelberg–New York 1981
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.