On Robin's criterion for the Riemann Hypothesis
arXiv:math/0604314
Abstract
Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality sum_{d|n}d<e^{gamma}n loglog n is satisfied for n>=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.
15 pages. Corrected version. In the first version in Theorem 5 (main result) it was falsely asserted that n must be superabundant, invalidating the proof. An alternative proof is provided in this version, some typos have been corrected, and the presentation has been improved