Farey sequences
type: Item
instance of: scholarly article
title: Farey sequences
zbMATH Open document ID: 0924.11008
DOI: 10.1007/BF02355009
author: S. B. Stechkin
published in: Mathematical Notes
publication date: 1999-01-11T00:00:00Z
review text: The Farey sequence \((h_1(n),\dots, h_{\Phi(n)} (n))\) consists of the members of \(H_n= \{\frac ab: a,b\in \mathbb{N}\), \(1\leq a\leq b\), \((a,b)=1\}\) arranged in increasing order. Here, \(\Phi(n)= | H_n| = \sum_{m\leq n}\varphi(m)\), where \(\varphi\) is Euler's function. For fixed \(n\), let \(\delta_i= \delta_i(n)= h_i(n)- \frac{i}{\Phi(n)}\); define \(S_p(n)= (\sum_i| \delta_i^p|)^{1/p}\) and \(\Delta(n)= \max\{| \sum_{i=1}^m \delta_i|: 1\leq m\leq \Phi(n)\}\). Both \(S_p(n)\) and \(\Delta(n)\) are related to the Riemannian hypothesis. The author gives various estimates, in particular lower bounds, for both \(S_p(n)\) and \(\Delta(n)\), some of them depending on \(\theta= \sup\{\sigma: \xi(\sigma+i_t)=0\}\). One of his bounds, namely \(S_2(n)\geq Cn^{-1/2} (\log n)^{1/2}\) shows the falsity of a conjecture of Sato, that \(S_2(n)= O(n^{-1/2}\log \log \log n)\).
Mathematics Subject Classification ID: 11B57
zbMATH DE Number: 1237982
zbMATH Keywords: Tauberian theorems
zbMATH Keywords: Farey sequence
zbMATH Keywords: Riemannian hypothesis
zbMATH Keywords: lower bounds
MaRDI profile type: Publication
cites work: Q5514658
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cites work: Rate of divergence of some integrals
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full work available at URL: https://doi.org/10.1007/bf02355009
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description: scientific article; zbMATH DE number 1237982