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

https://projecteuclid.org/euclid.em/1175789744

Mathematical Reviews number (MathSciNet)
MR2253548

Zentralblatt MATH identifier
1149.11041

Citation

Briggs, Keith. Abundant Numbers and the Riemann Hypothesis. Experiment. Math. 15 (2006), no. 2, 251--256. https://projecteuclid.org/euclid.em/1175789744