Preprint Short Note Version 7 Preserved in Portico This version is not peer-reviewed

# Properties of the Robin's Inequality

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. Properties of the Robin's Inequality. Preprints 2020, 2020020379 (doi: 10.20944/preprints202002.0379.v7). Vega, F. Properties of the Robin's Inequality. 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 $\frac{1}{2}$. Many consider it to be the most important unsolved problem in pure mathematics. The Robin's inequality consists in $\sigma(n) < e^{\gamma } \times n \times \ln \ln n$ where $\sigma(n)$ is the divisor function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Robin's inequality is true for every natural number $n > 5040$ if and only if the Riemann hypothesis is true. We prove the Robin's inequality is true for every natural number $n > 5040$ when $n$ is not divisible by any prime number $q_{m} \leq 113$. In addition, the Robin's inequality is true for every natural number $n = 113^{k} \times n' > 5040$ over an integer $k \geq 1$ when $(\ln n')^{\beta} \leq \ln n$, such that $\beta = \frac{113}{112}$ and $n'$ is not divisible by $113$.

## Subject Areas

number theory; inequality; divisor; prime

Comment 1
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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