On Nicolas criterion for the Riemann Hypothesis

Abstract

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 Cram\'er conjecture on gaps between consecutive primes. An analogous situation holds when replacing Euler totient by Dedekind $\Psi$ function.