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On Robin's criterion for the Riemann Hypothesis

Y. -J. Choie·N. Lichiardopol·P. Moree·P. Sole
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Abstract

Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality sum_{d|n}d =5041, where gamma denotes the Euler(-Mascheroni) constant. We show by elementary methods that if n>=37 does not satisfy Robin's criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that n must be divisible by a fifth power >1. As a consequence we infer that RH holds true if and only if every natural number divisible by a fifth power >1 satisfies Robin's inequality.

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Y. -J. Choie, N. Lichiardopol, P. Moree, P. Sole. 2006-09-07. On Robin's criterion for the Riemann Hypothesis. https://arxiv.org/abs/math/0604314

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