Superabundant numbers, their subsequences and the Riemann hypothesis
arXiv:1211.2147
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^γn\log\log 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.
This is an updated revised version of the article. arXiv admin note: text overlap with arXiv:1112.6010 by other authors