The Riemann Hypothesis

EasyChair Preprint no. 3708, version 46

4 pagesDate: April 27, 2021

Abstract

The Nicolas' theorem states that the Riemann Hypothesis is true if and only if the inequality $\prod_{q \mid p\#} \frac{q}{q-1} > e^{\gamma} \times \log\log p\#$ is true, where $p\#$ is a primorial for $p > 2$ and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. This means that the Nicolas' inequality is true when
$\liminf_{p\to \infty }(\frac{\prod_{q \mid p\#} \frac{q}{q-1}}{e^{\gamma} \times \log\log p\#})> 1.$
This is because of the Nicolas' theorem also states that the Riemann Hypothesis is false if and only if there are infinitely many primorial numbers for which the Nicolas' inequality is false and infinitely many others for which the Nicolas' inequality is true. However, we prove that
$\lim_{{p\to \infty }}(\frac{\prod_{q \mid p\#} \frac{q}{q-1}}{e^{\gamma} \times \log\log p\#})=1.$
In this way, we show that the Nicolas' inequality is false for many primorial numbers $p\#$ when $p$ tends to the infinity and thus, the Riemann Hypothesis is false too.

Keyphrases: Divisor, inequality, number theory