Entropy and Its Application to Number Theory

Seiji Fujino

doi:10.20944/preprints202203.0371.v3

Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed

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)

In this paper, we propose an extension of the Planck distribution function derived from the Boltzmann principle. That is, we consider extending Planck's law with new distribution functions. In addition, using ideas applied to the expansion of the Planck distribution function, Von Koch's inequality is derived without using the Riemann hypothesis, showing that the abc conjecture is negated. We also describe some challenges for the future. Namely, we will discuss that Entropy is related to dynamical systems described by logistic function models, such as the bacterial and the population growth.

Entropy; Boltzmann principle; Planck’s law; Dynamical system; Von Koch’s inequality; Riemann Hypothesis; abc conjecture

Physical Sciences, Thermodynamics

Comment 1

Received: 17 November 2023
Commenter: Seiji Fujino
Commenter's Conflict of Interests: Author

Comment: Correction points
1, Corrections to sentences and typos , Addition of references,
2, In section 3), modified the definition $W_{\pi_{f},x}$ ,
3, In subsection 3.3), corrected the transformation of the formula $S'_{\pi_f}(x)$,
4, In section 5), Changed the title of section 5) from {Issuse for the future} to {Generalization and application to dynamical systems}.
5, In section 5), modifyied the function $Q_D(x)$ to $Q_D(x)/\xi$ in $S_{D}(x)$, $S'_{D}(x)$ and $S''_{D}(x)$ .
Thanks a lot.
6, In section 5), deleted the discussion of Coulomb's law and Newton's law.

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Entropy and Its Application to Number Theory

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