Superabundant numbers, their subsequences and the Riemann hypothesis
Abstract
Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality \sigma(n)<e^\gamma n\log\log n for all n>5040 (\gamma is Euler's constant). It is a natural question in this direction to find a first integer, if exists, which violates this inequality. Following this process, we introduce a new sequence of numbers and call it as extremely abundant numbers. In this paper we show that the Riemann hypothesis is true, if and only if, there are infinitely many of these numbers. Moreover, we investigate some of their properties together with superabundant and colossally abundant numbers.
Cite
@article{arxiv.1211.2147,
title = {Superabundant numbers, their subsequences and the Riemann hypothesis},
author = {Sadegh Nazardonyavi and Semyon Yakubovich},
journal= {arXiv preprint arXiv:1211.2147},
year = {2013}
}
Comments
This is an updated revised version of the article. arXiv admin note: text overlap with arXiv:1112.6010 by other authors