On Robin’s criterion for the Riemann hypothesis

Abstract

Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality σ(n): = Σd|n d < eγn 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.