On Solé and Planat Criterion for the Riemann Hypothesis

EasyChair Preprint no. 10519, version 2

Abstract

There are several statements equivalent to the famous Riemann hypothesis. In 2011, Sol{\'e} 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 Sol{\'e} and Planat criterion, we prove that the Riemann hypothesis is true.

Keyphrases: Chebyshev function, prime numbers, Riemann hypothesis, Riemann zeta function

@Booklet{EasyChair:10519,
  author = {Frank Vega},
  title = {On Solé and Planat Criterion for the Riemann Hypothesis},
  howpublished = {EasyChair Preprint no. 10519},

  year = {EasyChair, 2023}}