## Experimental Mathematics

### Abundant Numbers and the Riemann Hypothesis

Keith Briggs

#### Abstract

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 information

Source
Experiment. Math., Volume 15, Issue 2 (2006), 251-256.

Dates
First available in Project Euclid: 5 April 2007