Square-free values of the Carmichael function.
type: Item
instance of: scholarly article
title: Square-free values of the Carmichael function.
zbMATH Open document ID: 1042.11058
DOI: 10.1016/S0022-314X(03)00110-0
author: Francesco Pappalardi
author: F. Saidak
author: Igor E. Shparlinski
published in: Journal of Number Theory
publication date: 2003-12-03T00:00:00Z
review text: For a positive integer \(n\) the Carmichael function \(\lambda(n)\) is the exponent of the multiplicative group modulo \(n\). In this paper, the authors find an asymptotic formula for the number \(L(x)\), which counts, for a given positive real number \(x\), the number of positive integers \(n\leq x\) having the property that \(\lambda(n)\) is square-free. Their result is that \[ L(x)\approx \kappa {{x}\over {\log^{1-\alpha} x}},\tag{1} \] where \(\alpha=\prod_{p\geq 2} (1-1/(p(p-1)))\) is the Artin constant and \(\kappa\) is a more complicated constant (explicitly determined). For the proof, the authors first use the Bombieri-Vinogradov theorem to show that if \(\pi_{sf}(x)\) counts the number of primes \(p\leq x\) with \(p-1\) square-free, then \(\pi_{sf}(x)=\alpha\pi(x)+O(x/\log^A x)\), where \(A>0\) is an arbitrary constant. Once equipped with this fact, the main result follows from the observation that the characteristic function of the set of those \(n\) such that \(\lambda(n)\) is square-free is multiplicative and supported on \(\pi_{sf}\), together with a well-known result of Wirsing relating the average value of a ``small multiplicative function'' \(f(n)\) to \(\sum_{p\leq x} f(p)\). The paper contains also some other interesting facts, such as a uniform upper bound for the function \(L\) in short intervals, a formula for the counting function of those positive integers \(n\) whose Carmichael function is \(k\)-free (for a fixed integer \(k\geq 2\)), as well as the results of some computations analyzing the speed of convergence to \(1\) of the ratio between \(L(x)\) and the right hand side of (1).
reviewed by: Florian Luca
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 2012767
zbMATH Keywords: square-free integer
zbMATH Keywords: Carmichael function
zbMATH Keywords: Wirsing theorem
MaRDI profile type: Publication
cites work: Q3836760
cites work: Q3900124
cites work: Primes in short arithmetic progressions with rapidly increasing differences
cites work: Q5752651
cites work: On the normal number of prime factors of \(\phi(n)\)
cites work: Carmichael's lambda function
cites work: Period of the power generator and small values of Carmichael's function
cites work: Q4052222
cites work: On Artin's conjecture.
cites work: On generalizing Artins conjecture on primitive roots to composite moduli
cites work: On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions \(\varphi\) and \(\sigma\)
cites work: The Number of Representations of an Integer as the Sum of a Prime and a k-Free Integer
cites work: Non-Abelian Generalizations of the Erdős-Kac Theorem
cites work: Q4338229
cites work: Q4737496
cites work: Q3248054
cites work: Das asymptotische Verhalten von Summen über multiplikative Funktionen
description: scientific article; zbMATH DE number 2012767