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

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.