Bemerkungen über Fordkugeln. (Remarks on Ford balls)
type: Item
instance of: scholarly article
title: Bemerkungen über Fordkugeln. (Remarks on Ford balls)
zbMATH Open document ID: 0616.10030
DOI: 10.1007/BF02941519
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1986-01-01T00:00:00Z
review text: For Gaussian integers a,b denote by F(a/b) the Ford ball, i.e. the closed ball in \({\mathbb{C}}\times {\mathbb{R}}\) with center (a/b, 1/2\(| b|^ 2)\) and radius \(1/2| b|^ 2\). \textit{G. J. Rieger} [J. Reine Angew. Math. 303/304, 1-20 (1978; Zbl 0392.10031)] studied the density of the Ford balls near the plane \({\mathbb{C}}\times \{0\}\), more precisely the area m(x) of the intersection of all Ford balls with the unit square in \({\mathbb{C}}\), shifted to height 1/x. For \(x\to \infty\) m(x) tends to 3/(4L(2,\(\chi)\)), \(\chi\) the nonprincipal character mod 4. In the present paper the error term m(x)-3/(4L(2,\(\chi)\)) is sharpened to \[ O(x^{- 1}\quad \exp (-c(\log x)^{3/5}\quad (\log \log x)^{-1/5})) \] resp. assuming a slightly generalized Riemann hypothesis to \(O(x^{- 13/12+\epsilon}).\) Further the author gives an asymptotic formula for the integral mean of the error term. The proofs utilize calculations for the Euler and Möbius function in algebraic number fields. The results can be extended to imaginary quadratic number fields with class number one.
Mathematics Subject Classification ID: 11J99
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11H99
zbMATH DE Number: 3997915
zbMATH Keywords: area of intersection
zbMATH Keywords: Ford ball
zbMATH Keywords: density
zbMATH Keywords: asymptotic formula
author: Werner Georg Nowak
reviewed by: Thomas Maxsein
MaRDI profile type: Publication
cites work: Fractions
cites work: Q3950619
cites work: On the prime ideal theorem
cites work: Q4402415
cites work: On Euler's \(\varphi\)-function in quadratic number fields
cites work: Über Ford-Kugeln.
cites work: Q5803461
full work available at URL: https://doi.org/10.1007/bf02941519
OpenAlex ID: W2316290324
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description: scientific article; zbMATH DE number 3997915