##
**
Extremely Abundant Numbers and the Riemann Hypothesis
**

###
Sadegh Nazardonyavi and Semyon Yakubovich

Departamento de Matemática

Faculdade de Ciências

Universidade do Porto

4169-007 Porto

Portugal

**Abstract:**

Robin's theorem states that the Riemann hypothesis is equivalent to the
inequality σ(*n*) < *e*^{γ} *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.

**
Full version: pdf,
dvi,
ps,
latex
**

(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.

Return to
**Journal of Integer Sequences home page**