Note on the Riemann Hypothesis

EasyChair Preprint 7520, version 14

Abstract

The Riemann Hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. In 2011, Sol{\'e} and and Planat stated that the Riemann Hypothesis is true if and only if the inequality $\prod_{q\leq q_{n}}\left(1+\frac{1}{q} \right) >\frac{e^{\gamma}}{\zeta(2)}\times \log\theta(q_{n})$ 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, Dedekind function, Riemann hypothesis, Riemann zeta function, prime numbers

@booklet{EasyChair:7520,
  author    = {Frank Vega},
  title     = {Note on the Riemann Hypothesis},
  howpublished = {EasyChair Preprint 7520},
  year      = {EasyChair, 2022}}