Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed

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)

How to cite: Vega, F. Disproof of the Riemann Hypothesis. Preprints 2021, 2021090480 (doi: 10.20944/preprints202109.0480.v3). Vega, F. Disproof of the Riemann Hypothesis. Preprints 2021, 2021090480 (doi: 10.20944/preprints202109.0480.v3).


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 found the first zero $y$ of the function $\upsilon(y)$ in $y \approx 8.2639316883312400623766461031726662911 \ E5565708$ for $C \geq 1$. In this way, we claim that the Riemann hypothesis could be false.


Riemann hypothesis; Nicolas inequality; Chebyshev function; prime numbers



Comments (1)

Comment 1
Received: 12 October 2021
Commenter: Frank Vega
Commenter's Conflict of Interests: Author
Comment: The previous version was almost fine, but there was an error calculating with the tool of the project PARI/GP since this uses an small precision of significant digits by default. Now, in this version we calculate the zero of the function which is approximatelly closer to "8.2639316883312400623766461031726662911   E5565708" for C = 1 with the function "solvestep" of the project PARI/GP that finds the zero of a function into an interval with a better accuracy (we use the interval [1000000!, 5000000!]). I changed the abstract, keywords and content.
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