An analogue of the Robin inequality of the second type for odd integers

TL;DR

This paper presents a new inequality related to the sum-of-divisors function for odd integers, extending Robin's inequality and providing bounds that could have implications for number theory conjectures.

Contribution

It introduces a variant of Robin's inequality specifically for odd integers, offering a new perspective on divisor sum bounds for this class of numbers.

Findings

Established a new inequality for odd integers involving the sum-of-divisors function

Provided explicit bounds with constants for the inequality

Extended the understanding of divisor sum inequalities for odd numbers

Abstract

In this paper we give a variant of the Robin inequality which states that $\frac{\sigma(n)}{n} \leq \frac{e^\gamma}{2} \log\log n + \frac{0.7398\cdots}{\log\log n}$ for any odd integer $n \geq 3$.