On the local behavior of the Carmichael \(\lambda\)-function
type: Item
instance of: scholarly article
title: On the local behavior of the Carmichael \(\lambda\)-function
zbMATH Open document ID: 1112.11047
author: Nicolas Doyon
author: Florian Luca
published in: Michigan Mathematical Journal
publication date: 2007-04-05T00:00:00Z
review text: Let \(\varphi\) be Euler's function, and let \(\lambda\) be Carmichael's function. Theorem 1. Let \(t\) be a positive integer and let \(a\) be any permutation of the set \(\{1,2,\ldots,t\}.\) There exist infinitely many positive integers \(n\) such that the inequality \(\lambda(n+a(i))>\lambda(n+a(i+1))\) holds for all \(i=1,\dots,t-1.\) Furthermore, if \(n:=n(t)\) denotes the minimal value of \(n\) such that the preceding inequality holds, then the estimate \(t>>\log_2n(t)\) holds as \(t\) tends to infinity. Theorem 2. Let \(t\) be a positive integer and let \(a\) and \(b\) be any permutations of the set \(\{1,2,\dots,t\}.\) There exist infinitely many positive integers \(n\) such that the inequalities \(\lambda(n+a(i))>\lambda(n+a(i+1))\) and \(\varphi(n+b(i))>\phi(n+b(i+1))\) hold for all \(i=1,\dots,t-1.\) Furthermore, if \(n:=n(t)\) denotes the minimal value of \(n\) such that the preceding inequalities hold, then the estimate \(t>>\left(\frac{\log_3n(t)}{\log_5n(t)}\right)^{\frac{1}{2}}\) holds as \(t\) tends to infinity.
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 5140632
zbMATH Keywords: Carmichael's function
zbMATH Keywords: Euler's function
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1156345594
OpenAlex ID: W2064513779
cites work: Q3836760
cites work: Q5752651
cites work: Q3900126
cites work: Zero-Free Regions for Dirichlet L-Functions, and the Least Prime in an Arithmetic Progression
cites work: The large sieve
cites work: Q3679282
DOI: 10.1307/MMJ/1156345594
description: scientific article; zbMATH DE number 5140632