Note on the Riemann Hypothesis

EasyChair Preprint no. 7520, version 11

8 pagesDate: May 17, 2022

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 Dedekind 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. We can deduce from that paper, if the Riemann Hypothesis is false, then the Dedekind inequality is not satisfied for infinitely many prime numbers $q_{n}$. Using this result, we prove the Riemann Hypothesis is true when $(1-\frac{0.15}{\log^{3}x})^{\frac{1}{x}}\times x^{\frac{1}{x}}\geq 1+\frac{\log(1-\frac{0.15}{\log^{3}x})+\log x}{x}$ is always satisfied for every sufficiently large positive number $x$. However, we know that inequality is trivially satisfied for every sufficiently large positive number $x$. In this way, we show the Riemann Hypothesis is true.

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