On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions ϕ and σ

Florian Luca, and Carl Pomerance. "On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions ϕ and σ." Colloquium Mathematicae 92.1 (2002): 111-130. <http://eudml.org/doc/283450>.

@article{FlorianLuca2002,
abstract = {Let σ(n) denote the sum of positive divisors of the integer n, and let ϕ denote Euler's function, that is, ϕ(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 σ(ϕ(n))/n ≥ 1/2 for all n. We show that σ(ϕ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(ϕ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that ϕ(n-ϕ(n)) < ϕ(n) on a set of asymptotic density 1.},
author = {Florian Luca, Carl Pomerance},
journal = {Colloquium Mathematicae},
keywords = {Euler's phi-function; sum-of-divisors function; maximal order; normal order; average order},
language = {eng},
number = {1},
pages = {111-130},
title = {On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions ϕ and σ},
url = {http://eudml.org/doc/283450},
volume = {92},
year = {2002},
}

TY - JOUR
AU - Florian Luca
AU - Carl Pomerance
TI - On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions ϕ and σ
JO - Colloquium Mathematicae
PY - 2002
VL - 92
IS - 1
SP - 111
EP - 130
AB - Let σ(n) denote the sum of positive divisors of the integer n, and let ϕ denote Euler's function, that is, ϕ(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 σ(ϕ(n))/n ≥ 1/2 for all n. We show that σ(ϕ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(ϕ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that ϕ(n-ϕ(n)) < ϕ(n) on a set of asymptotic density 1.
LA - eng
KW - Euler's phi-function; sum-of-divisors function; maximal order; normal order; average order
UR - http://eudml.org/doc/283450
ER -