Some effective inequalities between usual arithmetic functions
type: Item
instance of: scholarly article
title: Some effective inequalities between usual arithmetic functions
zbMATH Open document ID: 1232.11100
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2009-11-18T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1229696578
review text: Let \(\tau(n)\) and \(\sigma(n)\) be the number and the sum of the divisors of \(n\). Let \(\varphi\) denote the Euler function. Many effective upper bounds are derived for \(n/\varphi(n)\) in terms of \(\varphi(n)\) and for \(\sigma(n)/n\) in terms of \(\tau(n)\). For example, it is obtained that: \[ \begin{aligned} &\frac{n}{\varphi(n)}\leq \text{e}^{\gamma}\log\log\varphi(n)+3.65279\;\;\text{if}\;n\geq 3,\\ &\frac{n}{\varphi(n)}\leq 3\log\log\varphi(n)\;\;\text{if}\;n\geq 211,\\ &\frac{n}{\varphi(n)}\leq\text{e}^{\gamma}\log\log\varphi(n)+\frac{2.95504}{\log\log\varphi(n)}\;\;\text{if}\;n\geq 7,\\ &\frac{\sigma(n)}{n}\leq 2.59791\,\log\log(3\tau(n))\;\;\text{if}\;n\geq 2, \end{aligned} \] where the symbol \(\gamma\) denotes the Euler constant.
Mathematics Subject Classification ID: 11N56
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 5635555
zbMATH Keywords: Euler's function
zbMATH Keywords: sum of divisor function
zbMATH Keywords: champion numbers
zbMATH Keywords: highly composite numbers
author: Jean-Louis Nicolas
reviewed by: Jonas Šiaulys
MaRDI profile type: Publication
OpenAlex ID: W2065105375
DOI: 10.7169/FACM/1229696578
description: scientific article; zbMATH DE number 5635555