S ep 2 00 6 On Robin ’ s criterion for the Riemann Hypothesis

Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality σ(n) := d|n d < e γ n log log n is satisfied for n ≥ 5041, where γ 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 consequence we obtain that RH holds true iff every natural number divisible by a fifth power > 1 satisfies Robin's inequality.