On Dedekind Function 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 $\frac{1}{2}$. 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. On the one hand, the Robin's criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma} \cdot n \cdot \log \log n$ holds for all natural numbers $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function of $n$, $\gamma \approx 0.57721$ is the Euler-Mascheroni constant and $\log$ is the natural logarithm. Let $\Psi(n) = n \cdot \prod_{q \mid n} \left(1 + \frac{1}{q} \right)$ denote the Dedekind $\Psi$ function where $q \mid n$ means the prime $q$ divides $n$. We require the properties of primorial numbers, that is to say the primorial of order $n$ as $N_{n} = 2 \cdot \ldots \cdot q_{n}$. On the other hand, Sol{\'e} and Planat criterion states that the Riemann hypothesis is true if and only if the inequality $\zeta(2) \cdot \frac{\Psi(N_{n})}{N_{n}} > e^{\gamma} \cdot \log \theta(q_{n})$ holds for all prime numbers $q_{n}> 3$, where $\theta(x)$ is the Chebyshev function and $\zeta(x)$ is the Riemann zeta function. In this note, using both inequalities on primorial and prime numbers, we prove that the Riemann hypothesis is true.