Preprint Short Note Version 7 Preserved in Portico This version is not peer-reviewed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

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

In mathematics, 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 $\frac{1}{2}$. In 1915, Ramanujan proved that under the assumption of the Riemann Hypothesis, the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all sufficiently large $n$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. In 1984, Guy Robin proved that the inequality is true for all $n > 5040$ if and only if the Riemann Hypothesis is true. In 2002, Lagarias proved that if the inequality $\sigma(n) \leq H_{n} + exp(H_{n}) \times \log H_{n}$ holds for all $n \geq 1$, then the Riemann Hypothesis is true, where $H_{n}$ is the $n^{th}$ harmonic number. We show certain properties of these both inequalities that leave us to a proof of the Riemann Hypothesis.

Subject Areas

number theory; inequality; sum-of-divisors function; harmonic number; prime

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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