On numbers satisfying Robin's inequality, properties of the next counterexample and improved specific bounds

TL;DR

This paper investigates properties of numbers related to Robin's inequality, focusing on the next potential counterexample, and provides bounds and refinements for specific classes of numbers.

Contribution

It establishes new properties of the next Robin inequality counterexample and improves bounds for various number categories.

Findings

The next counterexample c is superabundant with over a billion prime divisors.

Bounds are derived for the ratio p_{ω(c)} / log c.

At most ω(c)/14 multiplicity parameters exceed 1.

Abstract

Define $s (n) := n^{- 1} \sigma (n)$ ($\sigma (n):=\sum_{d|n}d )$ and $\omega(n)$ is the number of prime divisors of $n$. One of the properties of $s$ plays a central role: $s (p^a) > s (q^b)$ if $p < q$ are prime numbers, with no special condition on $a, b$ other than $a, b \geqslant 1$. This result, combined with the Multiplicity Permutation theorem, will help us establish properties of the next counterexample (say $c$) to Robin's inequality $s (n) < e^{\gamma} \log \log n$. The number $c$ is superabundant, and $\omega(c)$ must be greater than a number close to one billion. In addition, the ratio $p_{\omega (c)} / \log c$ has a lower and upper bound. At most $\omega(c)/14$ multiplicity parameters are greater than $1$. Last but not least, we apply simple methods to sharpen Robin's inequality for various categories of numbers.