A Proof of the Riemann Hypothesis Through the Nicolas Inequality

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 $n$.