Journal of Integer Sequences, Vol. 17 (2014), Article 14.2.8

Extremely Abundant Numbers and the Riemann Hypothesis

Sadegh Nazardonyavi and Semyon Yakubovich
Departamento de Matemática
Faculdade de Ciências
Universidade do Porto
4169-007 Porto


Robin's theorem states that the Riemann hypothesis is equivalent to the inequality σ(n) < eγ n log log n for all n > 5040, where σ(n) is the sum of divisors of n and γ is Euler's constant. It is natural to seek the first integer, if it exists, that violates this inequality. We introduce the sequence of extremely abundant numbers, a subsequence of superabundant numbers, where one might look for this first violating integer. The Riemann hypothesis is true if and only if there are infinitely many extremely abundant numbers. These numbers have some connection to the colossally abundant numbers. We show the fragility of the Riemann hypothesis with respect to the terms of some supersets of extremely abundant numbers.

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(Concerned with sequences A004394 A004490 A217867.)

Received April 1 2013; revised versions received August 16 2013; January 15 2014. Published in Journal of Integer Sequences, February 7 2014.

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