On Solé and Planat Criterion for the Riemann Hypothesis

EasyChair Preprint 10519, version 17

Abstract

The Riemann hypothesis is the assertion that all non-trivial 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 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.

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

@booklet{EasyChair:10519,
  author    = {Frank Vega},
  title     = {On Solé and Planat Criterion for the Riemann Hypothesis},
  howpublished = {EasyChair Preprint 10519},
  year      = {EasyChair, 2023}}