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

Volume 92 / 2002

Florian Luca, Carl Pomerance Colloquium Mathematicum 92 (2002), 111-130 MSC: 11A25, 11N37, 11N56. DOI: 10.4064/cm92-1-10


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.


  • Florian LucaInstituto de Matemáticas de la UNAM
    Campus Morelia
    Ap. Postal 61-3 (Xangari)
    Morelia, Michoacán, Mexico
  • Carl PomeranceLucent Technologies Bell Laboratories
    600 Mountain Avenue
    Murray Hill, NJ 07974, U.S.A.

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