Reciprocal LCM powers and Robin's criterion

Abstract

For $\kappa >1$, 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 $\kappa$ and tends to ${\sigma }_{-1}$, 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 $1<\kappa \le 100,000$.

arXiv comment23 pages