A Proof of the Riemann Hypothesis Through the Nicolas Inequality

TL;DR

This paper proves that the Nicolas inequality, linked to the Riemann hypothesis, holds for all natural numbers, providing a potential proof of the Riemann hypothesis based on number theory and prime-related inequalities.

Contribution

It demonstrates that the Nicolas inequality is valid for all n, offering a new approach to prove the Riemann hypothesis.

Findings

Nicolas inequality holds for all n

Supports the truth of the Riemann hypothesis

Links between Mertens theorem and prime products

Abstract

A work by Nicolas has shown that if it can be proven that a certain inequality holds for all $n$, the Riemann hypothesis is true. This inequality is associated with the Mertens theorem, and hence the Euler totient at $\prod_{k=1}^n p_k$, where $n$ is any integer and $p_n$ is the $n$-th prime. We shall show that indeed the Nicolas inequality holds for all $n$.