Note on the Riemann Hypothesis

EasyChair Preprint no. 7520, version 3

4 pagesDate: March 17, 2022

Abstract

In mathematics, 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. We call this inequality as the Dedekind inequality. 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 argument, we prove the Riemann hypothesis is true when $\theta(q_{n})^{1 + \frac{1}{q_{n}}} \geq \theta(q_{n + 1})$ holds for a sufficiently large prime number $q_{n}$. We show this is equivalent to show that 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 $x$. Using the Puiseux series, we check by computer that $(1 - \frac{0.15}{\log^{3} x})^{\frac{1}{x}} \times x^{\frac{1}{x}}$ is $1 + \frac{\log(1 - \frac{0.15}{\log^{3} x})+ \log x}{x} + O\left(\left(\frac{1}{x} \right)^{2} \right)$ in the series expansion at $x = \infty$.

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