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Note for the Riemann Hypothesis
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: 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)
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Version 11 : Received: 23 June 2024 / Approved: 24 June 2024 / Online: 24 June 2024 (08:58:02 CEST)
Version 12 : Received: 25 June 2024 / Approved: 25 June 2024 / Online: 25 June 2024 (10:26:09 CEST)
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)
Version 10 : Received: 20 June 2024 / Approved: 21 June 2024 / Online: 21 June 2024 (10:43:53 CEST)
Version 11 : Received: 23 June 2024 / Approved: 24 June 2024 / Online: 24 June 2024 (08:58:02 CEST)
Version 12 : Received: 25 June 2024 / Approved: 25 June 2024 / Online: 25 June 2024 (10:26:09 CEST)
How to cite: Vega, F. Note for the Riemann Hypothesis. Preprints 2020, 2020020379. https://doi.org/10.20944/preprints202002.0379.v12 Vega, F. Note for the Riemann Hypothesis. Preprints 2020, 2020020379. https://doi.org/10.20944/preprints202002.0379.v12
Abstract
Let $\Psi(n) = n \cdot \prod_{q \mid n} \left(1 + \frac{1}{q} \right)$ denote the Dedekind $\Psi$ function where $q \mid n$ means the prime $q$ divides $n$. Define, for $n \geq 3$; the ratio $R(n) = \frac{\Psi(n)}{n \cdot \log \log n}$ where $\log$ is the natural logarithm. Let $N_{n} = 2 \cdot \ldots \cdot q_{n}$ be the primorial of order $n$. A trustworthy proof for the Riemann hypothesis has been considered as the Holy Grail of Mathematics by several authors. 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 $\frac{1}{2}$. There are several statements equivalent to the famous Riemann hypothesis. We show if the inequality $R(N_{n+1}) < R(N_{n})$ holds for $n$ big enough, then the Riemann hypothesis is true. In this note, we prove that $R(N_{n+1}) < R(N_{n})$ always holds for $n$ big enough.
Keywords
Riemann hypothesis; Prime numbers; Riemann zeta function; Chebyshev function
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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