On Nicolas criterion for the Riemann Hypothesis

Youngju Choie (1) , Michel Planat (2) , Patrick Solé (1)

1 Télécom ParisTech
2 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174)

Nicolas criterion for the Riemann Hypothesis is based on an inequality that Euler totient function must satisfy at primorial numbers. A natural approach to derive this inequality would be to prove that a specific sequence related to that bound is strictly decreasing. We show that, unfortunately, this latter fact would contradict Cramér conjecture on gaps between consecutive primes. An analogous situation holds when replacing Euler totient by Dedekind $\Psi$ function.

Mots clés: Nicolas inequality; Euler totient; Dedekind $\Psi$ function; Riemann Hypothesis; Primorial numbers

Domaines: Théorie des nombres [math.NT]

Youngju Choie, Michel Planat, Patrick Solé. On Nicolas criterion for the Riemann Hypothesis. 2010. ⟨hal-00546596v3⟩