Preprint Short Note Version 7 Preserved in Portico This version is not peer-reviewed

The Riemann Hypothesis

Version 1 : Received: 24 February 2020 / Approved: 25 February 2020 / Online: 25 February 2020 (12:21:49 CET)
Version 2 : Received: 27 February 2020 / Approved: 27 February 2020 / Online: 27 February 2020 (10:49:49 CET)
Version 3 : Received: 10 March 2020 / Approved: 11 March 2020 / Online: 11 March 2020 (16:04:28 CET)
Version 4 : Received: 31 March 2020 / Approved: 2 April 2020 / Online: 2 April 2020 (18:25:32 CEST)
Version 5 : Received: 20 April 2020 / Approved: 22 April 2020 / Online: 22 April 2020 (09:48:30 CEST)
Version 6 : Received: 3 June 2020 / Approved: 4 June 2020 / Online: 4 June 2020 (13:22:40 CEST)
Version 7 : Received: 6 June 2020 / Approved: 8 June 2020 / Online: 8 June 2020 (10:31:19 CEST)
Version 8 : Received: 2 July 2021 / Approved: 6 July 2021 / Online: 6 July 2021 (12:38:05 CEST)
Version 9 : Received: 14 October 2021 / Approved: 14 October 2021 / Online: 14 October 2021 (14:15:38 CEST)

How to cite: Vega, F. The Riemann Hypothesis. Preprints 2020, 2020020379 (doi: 10.20944/preprints202002.0379.v7). Vega, F. The Riemann Hypothesis. Preprints 2020, 2020020379 (doi: 10.20944/preprints202002.0379.v7).

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. It is known that $S(x)$ changes sign infinitely often. Using the Nicolas theorem, we prove that when the inequalities $\delta(x) \leq 0$ and $S(x) \geq 0$ are satisfied for some number $x \geq 127$, then the Riemann Hypothesis should be false. However, the Mertens second theorem states that $\lim_{{x\to \infty }} \delta(x) = 0$. Moreover, we know that $\lim_{{x\to \infty }} S(x) = 0$. In this way, this work could mean a new step forward in the direction for finally solving the Riemann Hypothesis.

Keywords

Riemann hypothesis; Nicolas theorem; prime numbers; Chebyshev function

Subject

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

Comments (1)

Comment 1
Received: 8 June 2020
Commenter: Frank Vega
Commenter's Conflict of Interests: Author
Comment: I have assumed in the previous version that the Goldbach's conjecture were true, because this won't have an infinite number of counterexamples. However, this assumption  have not been precisely proved yet. For that reason, we removed and changed the abstract and content of the paper.
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