On Solé and Planat criterion for the Riemann Hypothesis

Abstract

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 Euler-Mascheroni 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.