A disproof of the Riemann Hypothesis via the Nicolas Criterion

TL;DR

This paper claims to disprove the Riemann Hypothesis by refuting the Nicolas criterion through properties of related sequences and a contradiction argument.

Contribution

It provides a disproof of the Nicolas criterion for the Riemann Hypothesis by establishing properties of specific sequences and deriving a contradiction.

Findings

Nicolas inequality does not hold universally.

Properties of the product sequence are established.

Contradiction implies the Nicolas criterion is false.

Abstract

The achievement of this paper is a confutation of the inequality addressed by the Nicolas criterion for the Riemann Hypothesis, carried out after establishing properties of two related sequences. One of them is the product $\prod_{k=1}^n(1-1/p_k)$, rewritten as an alternating sum. The disproof is by contradiction: assuming the Nicolas inequality is always true, we reach an absurdity exploiting the aforementioned properties and a general lemma.