Download PDFOpen PDF in browserCurrent versionOn Solé and Planat Criterion for the Riemann HypothesisEasyChair Preprint no. 10519, version 1510 pages•Date: September 20, 2023AbstractThe Riemann hypothesis is the assertion that all nontrivial zeros have 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. In 2011, Solé and Planat stated that the Riemann hypothesis is true if and only if the inequality $\zeta(2) \cdot \prod_{q\leq q_{n}} (1+\frac{1}{q}) > e^{\gamma} \cdot \log \theta(q_{n})$ holds for all prime numbers $q_{n}> 3$, where $\theta(x)$ is the Chebyshev function, $\gamma \approx 0.57721$ is the EulerMascheroni constant, $\zeta(x)$ is the Riemann zeta function and $\log$ is the natural logarithm. In this note, using Solé and Planat criterion, we prove that the Riemann hypothesis is true. Keyphrases: Chebyshev function, prime numbers, Riemann hypothesis, Riemann zeta function
