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

Define $s(n):=n^{-1}\sigma (n)$ ($\sigma (n):={\sum }_{d\mathrm{\mid }n}d)$ 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 $a,b⩾1$. This result, combined with the Multiplicity Permutation theorem, will help us establish properties of the next counterexample (say $c$) to Robin’s inequality $s (n) < e^{\gamma} \log \log n$\mathrm{.}Thenumber$c$issuperabundant,and$\omega©$ must be greater than a number close to one billion. In addition, the ratio $p_{\omega (c)}\mathrm{/}\log c$ has a lower and upper bound. At most $\omega (c)\mathrm{/}14$ multiplicity parameters are greater than $1$. Last but not least, we apply simple methods to sharpen Robin’s inequality for various categories of numbers.