Répartition des valeurs de la fonction d'Euler. (Distribution of values of Euler's function)
type: Item
instance of: scholarly article
title: Répartition des valeurs de la fonction d'Euler. (Distribution of values of Euler's function)
zbMATH Open document ID: 0684.10002
author: Yanyan Li
published in: L'Enseignement Mathématique. 2e Série
publication date: 1989-01-01T00:00:00Z
review text: Denote by \(\phi\) Euler's function and let F(x) be the number of values of the \(\phi\)-function less than or equal to x. The best known result for the error term R(x) defined by \[ F(x)=\zeta (2)\zeta (3)/\zeta (6)\cdot x+R(x) \] is due to \textit{P. T. Bateman} [Acta Arith. 21, 329-345 (1972; Zbl 0217.319)]. Utilizing methods of complex analysis he showed \[ R(x)\quad \ll \quad x\cdot \exp (-(1-\epsilon)( \log x \log \log x)^{1/2}) \] for any \(\epsilon >0\). Using only elementary means \textit{J. L. Nicolas} [Enseign. Math., II. Ser. 30, 331-338 (1984; Zbl 0553.10036)] proved R(x) \(\ll x/\log x\). Generalizing this method the author improves the result to R(x) \(\ll x/\log^ 2 x.\) Reviewer's remark. The author conjectured that elementary methods lead to an estimation R(x) \(\ll x/\log^ k x\) for any \(k>0\). In the meantime \textit{M. Balazard} and the author have even proved \[ R(x)\quad \ll \quad x\cdot \exp (-c\sqrt{\log x}) \] which is very close to the above mentioned result of P. T. Bateman.
Mathematics Subject Classification ID: 11A25
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 4120232
zbMATH Keywords: values of Euler's \(\phi\)-function
zbMATH Keywords: asymptotic formulas
zbMATH Keywords: error term
reviewed by: Thomas Maxsein
MaRDI profile type: Publication
Recommended article: Distribution of values of Euler's functions
Recommended article: Q4378060
Recommended article: On the distribution of the values of Euler's function
Recommended article: Q5596958
Recommended article: Q5181769
Recommended article: Distribution des valeurs de la fonction d'Euler. (Distribution of the values of Euler's function)
Recommended article: Q4735243
Recommended article: Q5547690
Recommended article: Q4755013
Recommended article: On the distribution of some means concerning the Euler function
description: scientific article; zbMATH DE number 4120232