Large values of $n/\varphi (n)$ and $\sigma (n)/n$

Christian Axler, Jean-Louis Nicolas Acta Arithmetica MSC: Primary 11N56; Secondary 11N37. DOI: 10.4064/aa220705-12-10 Opublikowany online: 19 December 2022


Let $n$ be a positive integer, $\varphi (n)$ the Euler totient function, and $\sigma (n)=\sum _{d\mid n}d$ the sum of the divisors of $n$. It is easy to prove that $\sigma (n)/n\le n/\varphi (n)$. Landau proved that when $n\to \infty $, $\limsup n/(\varphi (n)\log \log n) = e^\gamma $, where $\gamma =0.577\ldots $ is the Euler constant, and a few years later, Gronwall proved that $\limsup \sigma (n)/(n\log \log n)$ is also equal to $e^\gamma $. Afterwards, several authors gave effective upper bounds for $n/\varphi (n)$ and $\sigma (n)/n$, either under the Riemann hypothesis or without assuming it. Let $X \ge 4$ be a real number and $\Phi (X)$ the maximum of $n/\varphi (n)$ for $n\le X$. Similarly, we denote by $\Sigma (X)$ the maximum of $\sigma (n)/n$ for $n\le X$. Our first result gives effective upper and lower bounds for $\Phi (X)/\Sigma (X)$. Next, we give new effective upper bounds for $n/\varphi (n)$ and for $\sigma (n)/n$.


  • Christian AxlerInstitute of Mathematics
    Heinrich Heine University Düsseldorf
    40225 Düsseldorf, Germany
  • Jean-Louis NicolasUniv. Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208
    Institut Camille Jordan, Mathématiques
    F-69622 Villeurbanne Cedex, France

Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Przepisz kod z obrazka

Odśwież obrazek

Odśwież obrazek