In this note I describe a computational study of the successive maxima of the relative sum-of-divisors function $\rho(n):=\sigma(n)/n$. These maxima occur at superabundant and colossally abundant numbers, and I also study the density of these numbers. The values are compared with the known maximal order $e^\gamma\loglog{n}$; theorems of Robin and Lagarias relate these data to a condition equivalent to the Riemann Hypothesis. It is thus interesting to see how close these conditions come to being violated.
@article{1175789744,
author = {Briggs, Keith},
title = {Abundant Numbers and the Riemann Hypothesis},
journal = {Experiment. Math.},
volume = {15},
number = {1},
year = {2006},
pages = { 251-256},
language = {en},
url = {http://dml.mathdoc.fr/item/1175789744}
}
Briggs, Keith. Abundant Numbers and the Riemann Hypothesis. Experiment. Math., Tome 15 (2006) no. 1, pp. 251-256. http://gdmltest.u-ga.fr/item/1175789744/