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Reciprocal LCM powers and Robin's criterion
Abstract
For κ> 1, we study the reciprocal LCM-power function σ[κ]−1(n) := ∑d | nμ(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≤ 100,000.
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BibTeX
@misc{molnar2026reciprocal,
title = {{Reciprocal LCM powers and Robin's criterion}},
author = {Grant Molnar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12816},
eprint = {2610.12816},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12816},
}
RIS
TY - UNPB
TI - Reciprocal LCM powers and Robin's criterion
AU - Grant Molnar
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12816
UR - https://arxiv.org/abs/2610.12816
AB - For κ> 1, we study the reciprocal LCM-power function σ^[κ]_−1(n) := ∑_d | nμ(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≤ 100,000.
ER -