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Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality $\sigma \left(n\right):={\sum }_{d|n}d\<e^{\gamma }nloglogn$ is satisfied for $n\ge 5041$, where $\gamma$ denotes the Euler(-Mascheroni) constant. We show by elementary methods that if $n\ge 37$ 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 consequence we obtain that RH holds true iff every natural number divisible by a fifth power...