On Solé and Planat criterion for the Riemann Hypothesis

Author:

Vega Frank

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.

Publisher

Qeios Ltd

Reference5 articles.

1. Raymond Ayoub. Euler and the zeta function. The American Mathematical Monthly, 81(10):1067–1086, 1974. doi:10.2307/2319041.

2. Patrick Solé and Michel Planat. Extreme values of the Dedekind ψ function. Journal of Combinatorics and Number Theory, 3(1):33–38, 2011.

3. Jean-Louis Nicolas. Petites valeurs de la fonction d’Euler. Journal of number theory, 17(3):375–388, 1983. doi:10.1016/0022-314X(83) 90055-0.

4. Franz Mertens. Ein Beitrag zur analytischen Zahlentheorie. J. reine angew. Math., 1874(78):46–62, 1874. doi:10.1515/crll.1874.78.46.

5. László Kozma. Useful Inequalities. http://www.lkozma.net/inequalities_cheat_sheet/ineq.pdf, 2023. Accessed 17 July 2023.

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