Superabundant numbers, their subsequences and the Riemann hypothesis

TL;DR

This paper introduces extremely abundant numbers and proves that the Riemann hypothesis holds if and only if infinitely many such numbers exist, linking number theory conjectures with new numerical sequences.

Contribution

The paper defines extremely abundant numbers and establishes their equivalence with the truth of the Riemann hypothesis, connecting new sequences to a major unsolved problem.

Findings

Riemann hypothesis is true iff infinitely many extremely abundant numbers exist.

Properties of extremely abundant numbers are analyzed alongside superabundant and colossally abundant numbers.

Introduces a new sequence related to divisor sum inequalities and number theory conjectures.

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.