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

• Autorzy: Florian Luca , Carl Pomerance
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11N37: Asymptotic results on arithmetic functions
• 11A25: Arithmetic functions; related numbers; inversion formulas
• 11N56: Rate of growth of arithmetic functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2002
• Tom: 92
• Numer: 1
• Strony: 111-130

Daty

wydano

2002

Twórcy

autor

Florian Luca

• Instituto de Matemáticas de la UNAM, Campus Morelia, Ap. Postal 61-3(Xangari), Morelia, Michoacán, Mexico

autor

Carl Pomerance

• Lucent Technologies Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, U.S.A.

Identyfikatory

DOI

10.4064/cm92-1-10