Reciprocal LCM powers and Robin's criterion

Source abstract

For $κ> 1$, we study the reciprocal LCM-power function $$σ^{[κ]}_{-1}(n) := \sum_{d \mid n}μ(d)\,σ_{-1}(n / d)^κ.$$ We determine its average and extremal orders. A natural Robin-scale normalization is strictly increasing in $κ$ and tends to $σ_{-1}$, yielding an RH-equivalent criterion. Unconditionally, the corresponding Robin inequality holds eventually on every bounded parameter interval. Directed interval arithmetic proves the inequality for each $n > 5040$ and $1 < κ\le 100{,}000$.