# Subsets of colossally abundant numbers

## TL;DR

This paper investigates the properties of colossally abundant numbers and their relation to Robin's hypothesis, establishing that Robin's inequality holds if and only if all CA2 subset numbers greater than 5040 satisfy it.

## Contribution

It classifies colossally abundant numbers into three subsets and proves the equivalence between Robin's hypothesis and the Robin inequality holding for CA2 numbers.

## Key findings

- Robin's hypothesis is true if and only if all CA2 numbers > 5040 satisfy Robin inequality.
- The paper provides a new classification of colossally abundant numbers into three disjoint subsets.
- The results offer a new perspective on the validity of Robin's hypothesis based on subset analysis.

## Abstract

Let $G(n)=\sigma (n)/(n \log \log n )$. Robin made hypothesis that $G(n)<e^\gamma$ for all integer $n>5040$. This article divides all colossally abundant numbers in to three disjoint subsets CA1, CA2 and CA3, and shows that Robin hypothesis is true if and only if all CA2 numbers $n>5040$ satisfy Robin inequality.