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 < en 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. YoungJu Choie Dept of Mathematics POSTECH Pohang, Korea 790-784 E-mail : [email protected] URL: http://www.postech.ac.kr/department/math/people/choieyoungju.htm Nicolas Lichiardopol ESSI Route des Colles 06 903 Sophia Antipolis, France E-mail : [email protected] Pieter Moree Max-Planck-Institut für Mathematik Manuscrit reçu le 19 juin 2006. 358 YoungJu Choie, Nicolas Lichiardopol, Pieter Moree, Patrick Solé Vivatsgasse 7 D-53111 Bonn, Germany E-mail : [email protected] URL: http://guests.mpim-bonn.mpg.de/moree/ Patrick Solé CNRS-I3S ESSI Route des Colles 06 903 Sophia Antipolis, France E-mail : [email protected] URL: http://www.i3s.unice.fr/~sole/