Superabundant numbers, their subsequences and the Riemann hypothesis

2012/11/09 by Sadegh Nazardonyavi, Semyon Yakubovich, Nazardonyavi, Sadegh +1
Mathematics · #11A25 #11K31 #11N37 #11Y70 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A25 #msc:11K31 #msc:11N37 #msc:11Y70

paper · pdf · doi:10.48550/arxiv.1211.2147

This is an updated revised version of the article. arXiv admin note: text overlap with arXiv:1112.6010 by other authors

arxiv created 2013/02/26 · arxiv updated 2013/02/27

Abstract

Let σ(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality σ(n)<e^γnloglog n for all n>5040 (γis 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.

Citations

• On SA, CA, and GA numbers

Related

• Delicacy of the Riemann hypothesis and certain subsequences of superabundant numbers
• On the Least Colossally Abundant Exception to Robin's Inequality
• A New Type of Abundant Numbers
• On Robin's inequality
• On numbers satisfying Robin's inequality, properties of the next counterexample and improved specific bounds