bibkey: choie2006robincriterion authors: Young-Ju Choie, Nicolas Lichiardopol, Pieter Moree, and Patrick Sole year: 2006 title: On Robin’s criterion for the Riemann Hypothesis doi: null url: https://arxiv.org/abs/math/0604314v2 claim: Robin’s strict inequality for every integer greater than 5040 is equivalent to the Riemann hypothesis; the reciprocal divisor sum is its equivalent left side. strata_touched: [] license: citation-only triage: anchor

On Robin’s criterion for the Riemann Hypothesis

The author version v2, abstract and §1, printed p.1, states Robin’s criterion as

$$\mathrm{RH}⟺\forall n\in \mathbb{N},n\ge 5041⟹\sigma (n)<e^{\gamma }n\log \log n.$$

Here $\gamma$ is the Euler–Mascheroni constant. The introduction also writes the left side after division by $n$ as ${\sum }_{d\mid n}1/d$. The authors attribute the equivalence to Robin, whose original paper is their reference [9]. This note consumes the criterion as recalled in v2, not a new proof of the criterion or of RH.

The finite divisor geometry of 5040 and any fixed prime support do not discharge the quantifier over all larger integers. No numerical Robin threshold, fifth-power reduction or global analytic error estimate is consumed by this note.