On Nicolas criterion for the Riemann Hypothesis

TL;DR

This paper investigates Nicolas criterion for the Riemann Hypothesis, showing that proving a key decreasing sequence would contradict Cramér's conjecture, thus highlighting the difficulty of establishing the criterion.

Contribution

It demonstrates that a natural approach to prove Nicolas criterion conflicts with Cramér's conjecture, revealing limitations in current methods.

Findings

Proving the decreasing sequence contradicts Cramér's conjecture.

Similar issues arise when replacing Euler totient with Dedekind Ψ function.

Highlights the challenge in proving Nicolas criterion via this approach.

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.