Entropy and Its Application to Number Theory

Seiji Fujino

doi:10.20944/preprints202203.0371.v2

Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed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 the expansion of the Planck distribution functions which is derived from the Boltzmann principle. Furthermore, we examine to expand Planck's law using new distribution functions. Moreover, using the ideas applied to the expansion of the Planck distribution function, we show that the derivation of Von Koch's inequality without using the Riemann Hypothesis and the negative consequence of the abc conjecture. Besides, we describe some issues for the future. Namely, we discuss that the Entropy is associated with the dynamical system, and the classical gravity theory of Newton's law and the electromagnetism of Coulomb's law by the law of inverse squares.

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

Physical Sciences, Thermodynamics

Comment 1

Received: 7 June 2022
Commenter: Seiji Fujino
Commenter's Conflict of Interests: Author

Comment: Entropy and Its application for Number Theory
Fixed as follows:

1. abstract and introduction,
2. Change $\tilde{R}$ before the formula (6.11) to $R^+_m$,
3. the formula (6.18) and changed \hat{R} to $R^-_m$,
4. bibliography.

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Entropy and Its Application to Number Theory

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