The Nicolas criterion for the Riemann Hypothesis

For every prime number $p_{n}$, we define the sequence $X_{n} = \prod_{q \leq p_{n}} \frac{q}{q-1} - e^{\gamma} \times \log \theta(p_{n})$, where $\theta(x)$ is the Chebyshev function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas theorem states that the Riemann hypothesis is true if and only if the $X_{n} > 0$ holds for all prime $p_{n} > 2$. For every prime number $p_{k}$, $X_{k} > 0$ is called the Nicolas inequality. We show if the sequence $X_{n}$ is strictly decreasing for $n$ big enough, then the Riemann hypothesis must be true. For every prime number $p_{n} > 2$, we define the sequence $Y_{n} = \frac{e^{\frac{1}{2 \times \log(p_{n})}}}{(1 - \frac{1}{\log(p_{n})})}$ and show that $Y_{n}$ is strictly decreasing for $p_{n} > 2$. For all prime $p_{n} > 286$, we demonstrate that the inequality $X_{n} < e^{\gamma} \times \log Y_{n}$ is always satisfied. We prove that $\lim_{{n\to \infty }}X_{n}=\lim_{{n\to \infty }}(\log Y_{n})=0$.