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Entropy and Its Application to Number Theory
Version 1
: Received: 27 March 2022 / Approved: 29 March 2022 / Online: 29 March 2022 (03:02:06 CEST)
Version 2 : Received: 6 June 2022 / Approved: 7 June 2022 / Online: 7 June 2022 (04:14:37 CEST)
Version 3 : Received: 16 November 2023 / Approved: 16 November 2023 / Online: 17 November 2023 (08:59:25 CET)
Version 4 : Received: 28 November 2023 / Approved: 29 November 2023 / Online: 29 November 2023 (10:59:25 CET)
Version 5 : Received: 18 December 2023 / Approved: 18 December 2023 / Online: 18 December 2023 (10:28:51 CET)
Version 6 : Received: 30 December 2023 / Approved: 30 December 2023 / Online: 30 December 2023 (16:24:49 CET)
Version 7 : Received: 8 January 2024 / Approved: 8 January 2024 / Online: 8 January 2024 (17:00:00 CET)
Version 2 : Received: 6 June 2022 / Approved: 7 June 2022 / Online: 7 June 2022 (04:14:37 CEST)
Version 3 : Received: 16 November 2023 / Approved: 16 November 2023 / Online: 17 November 2023 (08:59:25 CET)
Version 4 : Received: 28 November 2023 / Approved: 29 November 2023 / Online: 29 November 2023 (10:59:25 CET)
Version 5 : Received: 18 December 2023 / Approved: 18 December 2023 / Online: 18 December 2023 (10:28:51 CET)
Version 6 : Received: 30 December 2023 / Approved: 30 December 2023 / Online: 30 December 2023 (16:24:49 CET)
Version 7 : Received: 8 January 2024 / Approved: 8 January 2024 / Online: 8 January 2024 (17:00:00 CET)
How to cite: Fujino, S. Entropy and Its Application to Number Theory. Preprints 2022, 2022030371. https://doi.org/10.20944/preprints202203.0371.v4 Fujino, S. Entropy and Its Application to Number Theory. Preprints 2022, 2022030371. https://doi.org/10.20944/preprints202203.0371.v4
Abstract
In this paper, we propose an expansion of the Planck distribution function derived from the Boltzmann principle. Namely, we consider expanding Planck's law with a new distribution function, and discuss fine structure constant. Furthermore, using ideas applied to the expansion of the Planck distribution function, we show that Von Koch's inequality can be derived without using the Riemann Hypothesis, that is, the Riemann Hypothesis is true, and that the abc conjecture is negated. Furthermore, we define a generalization of Entropy and discuss that Entropy is relevant to dynamical systems described by logistic function models, such as the growth of bacteria or populations.
Keywords
Entropy; Boltzmann principle; Planck's law; Dynamical system; Logistic function; fine structure constant; Von Koch's inequality; Riemann Hypothesis; abc conjecture
Subject
Physical Sciences, Mathematical Physics
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: Seiji Fujino
Commenter's Conflict of Interests: Author
1, Abstract and Conclusion are modified,
2, In section 3), Integrate Lemma 3.6 and 3.7,
3, In subsection 3.3), modify the derivation of definition $n\pm(x,\alpha)$ and delete $S'_{\pi_f}(x)= Q'_f(x) \log (1+\frac{1}{Q_f(x)})$ .
4, In subsection 5.5) Added contents that logistic functions and dynamic systems are related entropy.
5, In section 2), Added about Planck's Radiation law. In this paper, to aid understanding,
we distinguish between the names Planck distribution function, Planck's law, and Planck's radiation law.
6, In section 3.4), and In subsection5.6) Added diagram,
7, In section 3.4), modified Defnition 3.10, Suggestion 3.11, and 3.12, because some issues.
8, Equation 4.33), fixed $\log (\frac{ \epsilon+2} {\epsilon+2})$ to $\log (\frac{ \epsilon+2} {\epsilon+1})$ ,
Thanks a lots.