Delicacy of the Riemann hypothesis and certain subsequences of superabundant numbers

TL;DR

This paper explores the delicate relationship between the Riemann hypothesis and specific subsets of superabundant numbers, highlighting how these subsets relate to Robin's criterion for RH.

Contribution

It demonstrates the sensitivity of Robin's theorem to certain subsequences of superabundant numbers, providing new insights into the structure related to RH.

Findings

RH is sensitive to subsets of superabundant numbers

Extremely abundant numbers are key to understanding Robin's criterion

Certain supersets of superabundant numbers influence the delicacy of RH

Abstract

Robin's theorem is one of the ingenious reformulation of the Riemann hypothesis (RH). It states that the RH is true if and only if $\sigma(n)<e^\gamma n\log\log n$ for all $n>5040$ where $\sigma(n)$ is the sum of divisors of $n$ and $\gamma$ is Euler's constant. In this paper we show that how the RH is delicate in terms of certain subsets of superabundant numbers, namely extremely abundant numbers and some of its specific supersets.