Preprint Article Version 4 Preserved in Portico This version is not peer-reviewed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 (doi: 10.20944/preprints202109.0480.v4). Vega, F. When the Riemann Hypothesis Might Be False. Preprints 2021, 2021090480 (doi: 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

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

Comment 1
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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