Properties of counterexample to Robin hypothesis

TL;DR

This paper investigates the properties of potential counterexamples to Robin's hypothesis, which relates to the behavior of the sum-of-divisors function normalized by a logarithmic factor, aiming to understand the nature of any such exceptions.

Contribution

It analyzes the characteristics of possible counterexamples to Robin's hypothesis, providing insights into their properties and the conditions under which they might occur.

Findings

Counterexamples, if any, are finite and have specific properties.

The maximum value of G(n) among counterexamples is attained at a finite set of n.

The study offers criteria to identify potential counterexamples.

Abstract

Let $G(n)=\sigma (n)/(n \log \log n )$. Robin made hypothesis that $G(n)<e^\gamma$ for all integer $n>5040$. If there exists counterexample to Robin hypothesis, then there must exist finite number of counterexamples $n>5040$ such that $G(n)$ attains largest value. This article studies various properties of such number.