## Experimental Mathematics

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

### 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 $\rho(n):=\sigma(n)/n$. 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

**Permanent link to this document**

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

**Mathematical Reviews number (MathSciNet)**

MR2253548

**Zentralblatt MATH identifier**

1149.11041

**Subjects**

Primary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses 11N64: Other results on the distribution of values or the characterization of arithmetic functions 11Y55: Calculation of integer sequences

**Keywords**

Riemann hypothesis abundant numbers

#### Citation

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