The Riemann Hypothesis

EasyChair Preprint no. 3708, version 52

6 pagesDate: July 15, 2021

Abstract

For every prime number $p_{n}$, we define the sequence $X_{n} = \frac{\prod_{q \mid N_{n}} \frac{q}{q-1}}{e^{\gamma} \times \log\log N_{n}}$, where $N_{n} = \prod_{k = 1}^{n} p_{k}$ is the primorial number of order $n$ and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas theorem states that the Riemann hypothesis is true if and only if the $X_{n} > 1$ holds for all prime $p_{n} > 2$. For every prime number $p_{k}$, $X_{k} > 1$ is called the Nicolas inequality. We show if the sequence $X_{n}$ is strictly decreasing for $n$ big enough, then the Riemann hypothesis should be true. Moreover, we demonstrate that the sequence $X_{n}$ is indeed strictly decreasing when $n \to \infty$. Notice that, Choie, Planat and Sol{\'e} in the preprint paper arXiv:1012.3613 have a proof that the Cram{\'e}r conjecture is false when $X_{n}$ is strictly decreasing for $n$ big enough. This paper is an extension of their result.

Keyphrases: Divisor, inequality, number theory