On some problems of M/akowski–Schinzel and Erdős
concerning the arithmetical functions $\phi $ and $\sigma $
Colloquium Mathematicum, Tome 92 (2002) no. 1, pp. 111-130
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences
Let $\sigma (n)$ denote the sum of positive divisors of the integer $n$, and let $\phi $ denote Euler's function, that is, $\phi (n)$ 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 $\sigma (\phi (n))/n\ge 1/2$ for all $n$. We show that $\sigma (\phi (n))/n\to \infty $ on a set of numbers $n$ of asymptotic density 1. In addition, we study the average order of $\sigma (\phi (n))/n$ as well as its range. We use similar methods to prove a conjecture of Erdős that $\phi (n-\phi (n))\phi (n)$ on a set of asymptotic density 1.
Keywords:
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
Affiliations des auteurs :
Florian Luca 1 ; Carl Pomerance 2
@article{10_4064_cm92_1_10,
author = {Florian Luca and Carl Pomerance},
title = {On some problems of {M/akowski{\textendash}Schinzel} and {Erd\H{o}s
concerning} the arithmetical functions $\phi $ and $\sigma $},
journal = {Colloquium Mathematicum},
pages = {111--130},
year = {2002},
volume = {92},
number = {1},
doi = {10.4064/cm92-1-10},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm92-1-10/}
}
TY - JOUR AU - Florian Luca AU - Carl Pomerance TI - On some problems of M/akowski–Schinzel and Erdős concerning the arithmetical functions $\phi $ and $\sigma $ JO - Colloquium Mathematicum PY - 2002 SP - 111 EP - 130 VL - 92 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4064/cm92-1-10/ DO - 10.4064/cm92-1-10 LA - en ID - 10_4064_cm92_1_10 ER -
%0 Journal Article %A Florian Luca %A Carl Pomerance %T On some problems of M/akowski–Schinzel and Erdős concerning the arithmetical functions $\phi $ and $\sigma $ %J Colloquium Mathematicum %D 2002 %P 111-130 %V 92 %N 1 %U http://geodesic.mathdoc.fr/articles/10.4064/cm92-1-10/ %R 10.4064/cm92-1-10 %G en %F 10_4064_cm92_1_10
Florian Luca; Carl Pomerance. On some problems of M/akowski–Schinzel and Erdős concerning the arithmetical functions $\phi $ and $\sigma $. Colloquium Mathematicum, Tome 92 (2002) no. 1, pp. 111-130. doi: 10.4064/cm92-1-10
Cité par Sources :