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# Disproof 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. Disproof of the Riemann Hypothesis. *Preprints* **2021**, 2021090480 (doi: 10.20944/preprints202109.0480.v2).
Vega, F. Disproof of the Riemann Hypothesis. Preprints 2021, 2021090480 (doi: 10.20944/preprints202109.0480.v2).

## Abstract

We define the function $\upsilon(x) = \frac{3 \times \log x + 5}{8 \times \pi \times \sqrt{x} + 1.2 \times \log x + 2} + \frac{\log x}{\log (x + C \times \sqrt{x} \times \log \log \log x)} - 1$ for some positive constant $C$ independent of $x$. We prove that the Riemann hypothesis is false when there exists some number $y \geq 13.1$ such that for all $x \geq y$ the inequality $\upsilon(x) \leq 0$ is always satisfied. We know that the function $\upsilon(x)$ is monotonically decreasing for all sufficiently large numbers $x \geq 13.1$. Hence, it is enough to find a value of $y \geq 13.1$ such that $\upsilon(y) \leq 0$ since for all $x \geq y$ we would have that $\upsilon(x) \leq \upsilon(y) \leq 0$. Using the tool $\textit{gp}$ from the project PARI/GP, we note that $\upsilon(100!) \approx \textit{-2.938735877055718770 E-39} < 0$ for all $C \geq \frac{1}{1000000!}$. In this way, we claim that the Riemann hypothesis could be false.

## Keywords

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

## Subject

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Commenter: Frank Vega

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