Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T18:49:42.274Z Has data issue: false hasContentIssue false
  • English
  • Français

On the number of integral ideals of given norm and ray-class

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  26 February 2010

R. W. K. Odoni
R. W. K. Odoni
Affiliation:
Prof. R. W. K. Odoni, Department of Mathematics, University of Glasgow, University Gardens, Glasgow, G12 8QW, Scotland.
Get access

Extract

Let K be an algebraic number field, [K: Q] = κ є N; only the case κ > 1 is of interest in this paper. Let f be any non-zero ideal in ZK, the ring of integers of K, and let b be any ray-class (modx f) of K. In this paper we answer a question of P. Erdös (private communication) about the “maximum-growth-rate” of the functions

and

the sum here taken over all ray-classes (modx f), while N(a) is the absolute norm of a. Let

and

where, as usual, for x є R, log+ x - log max {1, x}. We prove

THEOREM 1. For every f, b we have

MSC classification

Secondary: 11R45: Density theorems
Type
Research Article
Information
Mathematika , Volume 38 , Issue 1 , June 1991 , pp. 185 - 190
Copyright
Copyright © University College London 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Hardy, G. H. and Wright, E. M.. The Theory of Numbers 4th ed. (Oxford Univ. Press, 1960).Google Scholar
2. Hasse, H., Zahlbericht Tom I (Physica-Verlag, Wiirzburg, 1970).Google Scholar
3. Narkiewicz, W.. Elementary and Analytic Theory of Algebraic Numbers (PWN Warszawa, 1974).Google Scholar
4. Ramanujan, S.. Collected Papers, Papers 8 and 15 (Cambridge Univ. Press, 1927).Google Scholar
5. Wigert, S.. Arkiv. för matematik, 3 (1907), 1–9.Google Scholar