On some problems of M/akowski–Schinzel and Erdős concerning the arithmetical functions $\varphi$ and $\sigma$

Florian Luca¹ ; Carl Pomerance²

¹ Instituto de Matemáticas de la UNAM Campus Morelia Ap. Postal 61-3 (Xangari) Morelia, Michoacán, Mexico
² Lucent Technologies Bell Laboratories 600 Mountain Avenue Murray Hill, NJ 07974, U.S.A.

Colloquium Mathematicum, Tome 92 (2002) no. 1, pp. 111-130.

Résumé

Let $\sigma (n)$ denote the sum of positive divisors of the integer $n$, and let $\varphi$ denote Euler's function, that is, $\varphi (n)$ is the number of integers in the interval $[1,n]$ that are relatively prime to $n$. It has been conjectured by Mąkowski and Schinzel that $\sigma (\varphi (n))/n\ge 1/2$ for all $n$. We show that $\sigma (\varphi (n))/n\to \mathrm{\infty }$ on a set of numbers $n$ of asymptotic density 1. In addition, we study the average order of $\sigma (\varphi (n))/n$ as well as its range. We use similar methods to prove a conjecture of Erdős that $\varphi (n-\varphi (n))\varphi (n)$ on a set of asymptotic density 1.

Détail

DOI : 10.4064/cm92-1-10

Mots-clés : sigma denote sum positive divisors integer phi denote eulers function phi number integers interval relatively prime has conjectured kowski schinzel sigma phi sigma phi infty set numbers asymptotic density addition study average order sigma phi its range similar methods prove conjecture erd phi n phi phi set asymptotic density