On numbers satisfying Robin's inequality, properties of the next counterexample and improved specific bounds
英文摘要：
Define $s (n) := n^{- 1} σ(n)$ ($σ(n):=\sum_{d|n}d )$ and $ω(n)$ 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 $s (n) < e^γ \log \log n$. The number $c$ is superabundant, and $ω(c)$ must be greater than a number close to one billion. In addition, the ratio $p_{ω(c)} / \log c$ has a lower and upper bound. At most $ω(c)/14$ multiplicity parameters are greater than $1$. Last but not least, we apply simple methods to sharpen Robin's inequality for various categories of numbers.
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