Let us denote by $\tau(n)$ and $\sigma(n)$ the number and the sum of the divisors of $n$ and by
$\varphi$ Euler's function. We give effective upper bounds for $\frac{n}{\varphi(n)}$ in terms of $\varphi(n)$, and for $\frac{\sigma(n)}{n}$ in terms of $\tau(n)$.
Publié le : 2008-12-15
Classification:
Euler's function,
sum of divisors function,
champion numbers,
highly composite numbers,
11N56
@article{1229696578,
author = {Nicolas, Jean-Louis},
title = {Quelques in\'egalit\'es effectives entre des fonctions arithm\'etiques usuelles},
journal = {Funct. Approx. Comment. Math.},
volume = {38},
number = {1},
year = {2008},
pages = { 315-334},
language = {fr},
url = {http://dml.mathdoc.fr/item/1229696578}
}
Nicolas, Jean-Louis. Quelques inégalités effectives entre des fonctions arithmétiques usuelles. Funct. Approx. Comment. Math., Tome 38 (2008) no. 1, pp. 315-334. http://gdmltest.u-ga.fr/item/1229696578/