On Solé and Planat criterion for the Riemann hypothesis

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. This is one of the Clay Mathematics Institute's Millennium Prize Problems. There are several statements equivalent to the famous Riemann hypothesis. In 2011, Sol{\'e} and Planat stated that the Riemann hypothesis is true if and only if the inequality $\zeta(2) \cdot \prod_{p\leq p_{k}} (1+\frac{1}{p}) > e^{\gamma} \cdot \log \theta(p_{k})$ holds for all prime numbers $p_{k}> 3$, where $\theta(x)$ is the Chebyshev function, $\gamma \approx 0.57721$ is the Euler-Mascheroni constant, $\zeta(x)$ is the Riemann zeta function and $\log$ is the natural logarithm. Let $\sigma(n)$ denote the sum-of-divisors function of $n$. We also require the properties of superabundant numbers, that is to say left to right maxima of $n \mapsto \frac{\sigma(n)}{n}$. In this note, using Sol{\'e} and Planat criterion on superabundant numbers, we prove that the Riemann hypothesis is true.