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# The Complete Proof of the Riemann Hypothesis

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)

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. The Complete Proof of the Riemann Hypothesis. *Preprints* **2021**, 2021090480 (doi: 10.20944/preprints202109.0480.v1).
Vega, F. The Complete Proof of the Riemann Hypothesis. Preprints 2021, 2021090480 (doi: 10.20944/preprints202109.0480.v1).

## 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 $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. We prove that the Robin inequality is true for all $n > 5040$ which are not divisible by any prime number between $2$ and $953$. Using this result, we show there is a contradiction just assuming the possible smallest counterexample $n > 5040$ of the Robin inequality. In this way, we prove that the Robin inequality is true for all $n > 5040$ and thus, the Riemann Hypothesis is true.

## Keywords

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

## Subject

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

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