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When the Riemann Hypothesis Might Be False
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: Received: 27 September 2021 / Approved: 29 September 2021 / Online: 29 September 2021 (08:30:39 CEST)
Version 2 : Received: 11 October 2021 / Approved: 11 October 2021 / Online: 11 October 2021 (15:38:28 CEST)
Version 3 : Received: 12 October 2021 / Approved: 12 October 2021 / Online: 12 October 2021 (14:31:46 CEST)
Version 4 : Received: 14 October 2021 / Approved: 15 October 2021 / Online: 15 October 2021 (11:14:58 CEST)
Version 5 : Received: 22 July 2024 / Approved: 22 July 2024 / Online: 23 July 2024 (07:33:25 CEST)
Version 6 : Received: 23 July 2024 / Approved: 23 July 2024 / Online: 23 July 2024 (13:53:56 CEST)
Version 7 : Received: 12 August 2024 / Approved: 12 August 2024 / Online: 13 August 2024 (07:51:25 CEST)
Version 8 : Received: 19 August 2024 / Approved: 19 August 2024 / Online: 19 August 2024 (11:18:42 CEST)
Version 9 : Received: 25 September 2024 / Approved: 26 September 2024 / Online: 26 September 2024 (12:15:52 CEST)
Version 2 : Received: 11 October 2021 / Approved: 11 October 2021 / Online: 11 October 2021 (15:38:28 CEST)
Version 3 : Received: 12 October 2021 / Approved: 12 October 2021 / Online: 12 October 2021 (14:31:46 CEST)
Version 4 : Received: 14 October 2021 / Approved: 15 October 2021 / Online: 15 October 2021 (11:14:58 CEST)
Version 5 : Received: 22 July 2024 / Approved: 22 July 2024 / Online: 23 July 2024 (07:33:25 CEST)
Version 6 : Received: 23 July 2024 / Approved: 23 July 2024 / Online: 23 July 2024 (13:53:56 CEST)
Version 7 : Received: 12 August 2024 / Approved: 12 August 2024 / Online: 13 August 2024 (07:51:25 CEST)
Version 8 : Received: 19 August 2024 / Approved: 19 August 2024 / Online: 19 August 2024 (11:18:42 CEST)
Version 9 : Received: 25 September 2024 / Approved: 26 September 2024 / Online: 26 September 2024 (12:15:52 CEST)
How to cite: Vega, F. When the Riemann Hypothesis Might Be False. Preprints 2021, 2021090480. https://doi.org/10.20944/preprints202109.0480.v4 Vega, F. When the Riemann Hypothesis Might Be False. Preprints 2021, 2021090480. https://doi.org/10.20944/preprints202109.0480.v4
Abstract
Robin criterion states that the Riemann Hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. Let $q_{1} = 2, q_{2} = 3, \ldots, q_{m}$ denote the first $m$ consecutive primes, then an integer of the form $\prod_{i=1}^{m} q_{i}^{a_{i}}$ with $a_{1} \geq a_{2} \geq \cdots \geq a_{m} \geq 0$ is called an Hardy-Ramanujan integer. If the Riemann Hypothesis is false, then there are infinitely many Hardy-Ramanujan integers $n > 5040$ such that Robin inequality does not hold and $n < (4.48311)^{m} \times N_{m}$, where $N_{m} = \prod_{i = 1}^{m} q_{i}$ is the primorial number of order $m$.
Keywords
Riemann hypothesis; Robin inequality; sum-of-divisors function; prime numbers
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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