An interval result for the number field \(\psi{}(x,y)\) function
type: Item
instance of: scholarly article
title: An interval result for the number field \(\psi{}(x,y)\) function
zbMATH Open document ID: 0772.11044
DOI: 10.1007/BF02567771
author: Pieter Moree
published in: Manuscripta Mathematica
publication date: 1993-06-29T00:00:00Z
full work available at URL: https://eudml.org/doc/155764
review text: For a given number field \(K\), define \(\psi_ K(x,y)\) to be the number of ideals in the ring of integers of \(K\) with norm \(\leq x\),and free of prime ideal factors with norm \(>y\). This function has been well-studied for \(K=\mathbb{Q}\); and recent authors have shown that analogous methods carry over to arbitrary number fields. Here the author shows that one can get `sharp' asymptotic estimates for \(\psi_ K(cx,y)-\psi_ K(x,y)\), where \(1\leq c\leq y\), the range depending on the range in which one can give sharp estimates for the number of prime ideals whose norm lie in short intervals. The author's result is directly analogous to that of \textit{A. Hildebrand} for \(\mathbb{Q}\) [Trans. Am. Math. Soc. 297, 729-751 (1986; Zbl 0611.10028)].
Mathematics Subject Classification ID: 11R47
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11R44
zbMATH DE Number: 219913
zbMATH Keywords: number of ideals
zbMATH Keywords: asymptotic estimates
zbMATH Keywords: number of prime ideals
zbMATH Keywords: short intervals
reviewed by: Andrew Granville
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 219913