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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.
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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

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References

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