Criterion for the Riemann Hypothesis

Overview & Synopsis

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 $n$. Define, for $n \geq 3$; the ratio $R(n) = \frac{\Psi(n)}{n \cdot \log \log n}$ where $\log$ is the natural logarithm. Let $M_{x} = \prod_{q \leq x} q$ be the product extending over all prime numbers $q$ that are less than or equal to $x$. 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. We state that if the Riemann hypothesis is false, then there exist infinitely natural numbers $x$ such that the inequality $R(M_{x}) < \frac{e^{\gamma}}{\zeta(2)}$ holds, where $\gamma \approx 0.57721$ is the Euler-Mascheroni constant and $\zeta(x)$ is the Riemann zeta function. In this note, using our criterion, we prove that the Riemann hypothesis is true.