New Criterion for the Riemann Hypothesis

EasyChair Preprint 10352

Abstract

There are several statements equivalent to the famous Riemann hypothesis. In 2011, Sol{\'e} and and Planat stated that the Riemann hypothesis is true if and only if the inequality $\prod_{q\leq q_{n}}\left(1+\frac{1}{q} \right) >\frac{e^{\gamma}}{\zeta(2)}\cdot \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 3$, where $\theta(x)$ is the Chebyshev function, $\gamma\approx 0.57721$ is the Euler-Mascheroni constant and $\zeta(x)$ is the Riemann zeta function. Using this result, we create a new criterion for the Riemann hypothesis. We prove the Riemann hypothesis is true using this new criterion.

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

@booklet{EasyChair:10352,
  author    = {Frank Vega},
  title     = {New Criterion for the Riemann Hypothesis},
  howpublished = {EasyChair Preprint 10352},
  year      = {EasyChair, 2023}}