Preprint Article Version 4 Preserved in Portico This version is not peer-reviewed

When the Riemann Hypothesis Might Be False

Version 1 : 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)

How to cite: Vega, F. When the Riemann Hypothesis Might Be False. Preprints 2021, 2021090480. Vega, F. When the Riemann Hypothesis Might Be False. Preprints 2021, 2021090480.


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 $m$.


Riemann hypothesis; Robin inequality; sum-of-divisors function; prime numbers


Computer Science and Mathematics, Algebra and Number Theory

Comments (1)

Comment 1
Received: 15 October 2021
Commenter: Frank Vega
Commenter's Conflict of Interests: Author
Comment: The previous version were flawed. Now, I'm submitting a new version that is correct and it is not similar to previous submission. I changed the title, abstract, keywords and content.
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