Experimental Mathematics

Abundant Numbers and the Riemann Hypothesis

Keith Briggs

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


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