On Nicolas Criterion for the Riemann Hypothesis

The Riemann hypothesis is the assertion that all non-trivial zeros are complex numbers with real part $\frac{1}{2}$. It is considered by many to be the most important unsolved problem in pure mathematics. There are several statements equivalent to the famous Riemann hypothesis. For $x \geq 2$, the function $f$ was introduced by Nicolas in his seminal paper as $f(x) = e^{\gamma} \cdot \log\theta(x) \cdot \prod_{q \leq x} \left(1 - \frac{1}{q} \right)$, where $\theta(x)$ is the Chebyshev function, $\gamma \approx 0.57721$ is the Euler-Mascheroni constant and $\log$ is the natural logarithm. In 1983, Nicolas stated that if the Riemann hypothesis is false then there exists a real number $b$ with $0 < b < \frac{1}{2}$ such that, as $x\to \infty$, $\log f(x)=\Omega_{\pm }(x^{-b})$. In this note, using the Nicolas criterion, we prove that the Riemann hypothesis is true.