Entropy and Its Application to Number Theory

Seiji Fujino

doi:10.20944/preprints202203.0371.v7

Preprint Article Version 7 Preserved in Portico This version is not peer-reviewed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 suggest an expansion of the Planck distribution function derived from the Boltzmann principle. Thereby, 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 not true. 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 and populations.

Entropy; Boltzmann principle; Planck's law; Dynamical system; Logistic function; fine structure constant; Von Koch's inequality; Riemann Hypothesis; abc conjecture

Physical Sciences, Mathematical Physics

Comment 1

Received: 8 January 2024
Commenter: Seiji Fujino
Commenter's Conflict of Interests: Author

Comment: Fixed that some editing errors and omissions in the numbering of definitions, lemmas, theorems, corollaries, proposals, etc.
Thanks.

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Entropy and Its Application to Number Theory

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