Download PDFOpen PDF in browserCurrent versionNote on the Riemann HypothesisEasyChair Preprint 7520, version 58 pages•Date: April 3, 2022AbstractIn mathematics, 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 $\frac{1}{2}$. In 2011, Sol{\'e} and and Planat stated that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \left(1 + \frac{1}{q} \right) > \frac{e^{\gamma}}{\zeta(2)} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 3$, where $\theta(x)$ is the Chebyshev function, $\gamma \approx 0.57721$ is the EulerMascheroni constant and $\zeta(x)$ is the Riemann zeta function. We call this inequality the Dedekind inequality. We can deduce from that paper, if the Riemann hypothesis is false, then the Dedekind inequality is not satisfied for infinitely many prime numbers $q_{n}$. Using this argument, we prove the Riemann hypothesis is true when $\theta(q_{n})^{1 + \frac{1}{q_{n}}} \geq \theta(q_{n + 1})$ holds for a sufficiently large prime number $q_{n}$. We show this is equivalent to state that the Riemann hypothesis is true when $(1  \frac{0.15}{\log^{3} x})^{\frac{1}{x}} \times x^{\frac{1}{x}} \geq 1 + \frac{\log(1  \frac{0.15}{\log^{3} x})+ \log x}{x}$ is always satisfied for every sufficiently large positive number $x$. However, we know that inequality is trivially satisfied for every sufficiently large positive number $x$. In this way, we prove the Riemann hypothesis is true. Keyphrases: Chebyshev function, Dedekind function, Riemann hypothesis, Riemann zeta function, prime numbers
