A concentration property with respect to the Euler function
type: Item
instance of: scholarly article
title: A concentration property with respect to the Euler function
zbMATH Open document ID: 0963.11042
DOI: 10.1016/S0019-3577(98)80007-4
author: F. Marie-Jeanne
author: Gérald Tenenbaum
published in: Indagationes Mathematicae. New Series
publication date: 1999-06-01T00:00:00Z
review text: Let \(E(n)=\Omega (\varphi (n))\) and \[ Q(x,E) = x^{-1} \max _k \# \{n\leq x: E(n)=k \} . \] \textit{P. Erdős} and \textit{C. Pomerance} [Rocky Mt. J. Math. 15, 343-352 (1985; Zbl 0617.10037)] proved that \(E\) has a normal limiting distribution with norming \( ( \log\log x)^{3/2} \). This led \textit{M. Balazard} and \textit{A. Smáti} [Groupe de Travail en Théorie Analytique et Élémentaire des Nombres 1989-1990, Publ. Math. Orsay 92-01, 5-36 (1992; Zbl 0780.11005)] to conjecture that \(Q(x, E) \asymp (\log\log x)^{-3/2}\). This conjecture is established in the present paper. This requires rather elaborate study of this function since (1) the number-theoretical analogs of the Kolmorov-Rogozin inequality, as given by Halász and the reviewer, are principally uncapable to give a better estimate than \((\log\log x)^{-1/2}\), and (2) as the value of \(E(p)\) depends on multiplicative properties of \(p-1\), even the simple auxiliary properties of \(E\) are often difficult to establish.
Mathematics Subject Classification ID: 11K65
Mathematics Subject Classification ID: 11N60
zbMATH DE Number: 1251200
zbMATH Keywords: Euler's phi-function
zbMATH Keywords: concentration
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1251200