The Riemann Hypothesis

EasyChair Preprint no. 3708, version 48

5 pagesDate: May 10, 2021

Abstract

Let's define $\delta(x) = (\sum_{{q\leq x}}{\frac{1}{q}}-\log \log x-B)$, where $B \approx 0.2614972128$ is the Meissel-Mertens constant. The Robin theorem states that $\delta(x)$ changes sign infinitely often. Let's also define $S(x) = \theta(x) - x$, where $\theta(x)$ is the Chebyshev function. A theorem due to Erhard Schmidt implies that $S(x)$ changes sign infinitely often. Using the Nicolas theorem, we prove that when the inequalities $\delta(p) \leq 0$ and $S(p) \geq 0$ are satisfied for a prime $p \geq 127$, then the Riemann Hypothesis should be false. However, we could restate the Mertens second theorem as $\lim_{{n\to \infty }} \delta(p_{n}) = 0$ where $p_{n}$ is the $n^{th}$ prime number. In addition, we could modify the well-known formula $\lim_{{n \to \infty }} \frac{\theta(p_{n})}{p_{n}} = 1$ as $\lim_{{n\to \infty }} S(p_{n}) = 0$. In this way, this work could mean a new step forward in the direction for finally solving the Riemann Hypothesis.

Keyphrases: Divisor, inequality, number theory