Abundant Numbers and the Riemann Hypothesis

Abstract

In this note I describe a computational study of the successive maxima of the relative sum-of-divisors function ρ(n) := σ(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 eã log log (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.

Keywords:

• 11M26, 11N64, 11Y55
• Riemann hypothesis
• abundant numbers