Discrepancy of Farey sequences
type: Item
instance of: scholarly article
title: Discrepancy of Farey sequences
zbMATH Open document ID: 0981.11026
author: François Dress
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2000-04-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/225810
full work available at URL: jtnb11-2.html
review text: The author proves the very remarkable result, that the absolute discrepancy of the Farey sequence of order \(n\), consisting of all rational numbers in the unit interval of denominator smaller or equal to \(n\), is exactly \(1/n\). The right order of growth was already obtained in \textit{H. Niederreiter} [Math. Ann. 201, 341-345 (1973; Zbl 0248.10013)]. An analogous result for the square mean discrepancy is equivalent to the Riemann hypothesis by an old result of Franel. The proof given here is divided into two difficult parts \((n< 10^{110}\) and \(N> 10^{400})\), studies ``major and minor arcs'' and uses upper bounds of an integral related to a summatory function of the Möbius function. One has to estimate carefully several special cases, in particular in connection with rationals of small denominators. Useful is also a concept of truncated convergence, neglecting ``small relative errors'' (of order smaller than \(10^{-100})\).
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 11K31
zbMATH DE Number: 1426376
zbMATH Keywords: absolute discrepancy
zbMATH Keywords: Farey sequence of order \(n\)
reviewed by: Harald Rindler
MaRDI profile type: Publication
OpenAlex ID: W1971257223
cites work: Fonction sommatoire de la fonction de Möbius 1. Majorations expérimentales
cites work: Fonction sommatoire de la fonction de Mobius 2. Majorations asymptotiques élémentalres
cites work: Summatory function of the Möbius function. III: Strong effective asymptotic upper bounds
cites work: The distribution of Farey points
DOI: 10.5802/JTNB.255
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description: scientific article; zbMATH DE number 1426376