Dr. Jonathan Kenigson, FRSA (UK)

Analytic Number Theory (ANT) and Continuous Dynamical Systems make odd bedfellows. In the first instance, it may appear that questions regarding the distribution of primes over the integers are entirely unrelated to basic physical intuitions. Indeed, there is an extent to which this isometry has been discussed in terms of Renormalization Paradigms. One “begs the question” of “why Zetas and Gammas appear in physical paradigms” by using Hawking-Hartle paradigms to scale the paradigms. The difficult question – at least for the philosopher – is whether such paradigms possess predictive agency, or whether, perhaps, they are matters of convenience or pure mathematical suitability. The Statistical Mechanics of Boltzmann Distributions and of Fermi-Dirac Distributions are related by the quasi-physical intuition that the Central Limit Theorem (CLT) of Kolmogorov’s Measure-Theoretic Statistics should apply in diverse empirical contexts. Why, however, do Boltzmann and Fermi-Dirac and Bose-Einstein distributions dovetail so effortlessly in mathematical formalism as well as in practically-observable empirical contexts? For the Combinatorist, this “ubiquity question” has to do more with Counting Principles than with physical intuitions. Indeed, Fermionic and Bosonic distributions can be easily derived from state-occupancy arguments that are Monentum- and Position- agnostic. Heisenberg Theory – situated upon the aloof precipice of Operator Algebras and C* Algebras – encodes the “original intuition” that the founders of Quantum Mechanics nearly collectively possessed: That Nature operates according to Parametric Statistics and that Functional Analysis provides a convenient and rigorous generalization of the properties of State Spaces. One could wryly apply “Dead/Live Cat Arguments” to the question of whether “State Spaces…exist abstractly” or whether such spaces “exist apart from humans or their practices.” The Platonist, the Constructivist, the Structuralist, and the working mathematician are obliged to take a long backward glance and appreciate the origins of paradigms. Perhaps “State Spaces” exist in the same contingent sense as Schrodinger’s Cat, or could be problematized better physically – e.g. as a consequence of some paradigm that governs the contingency of observations in accordance with Fuzzy Logic.

This halting and possibly fruitless discussion can be superseded by the immediacy of results hard-won and far-afield. If Platonists are born and then destroyed, perhaps they are formed again, restless as ghosts in a Midsummer Night’s Dream. Perhaps to be thoroughly desolated by the Nominalist is to long again for a callow certitude that is not possible in light of philosophical maturity. If this is the case, then the “discoveries” of the working mathematician are specious. Whatever the case, one may still marvel at the implications of the chorus of ostensibly unrelated fields and admire – to the degree possible – either the art or casuistry of the conductor. I present one such marvelous case that to me inspires naive wonder and revives the scorned merriment and philistine hope of the youth who courted certainty and found scant little of it.

2022 (December): Consider a Riemmanian Magneton quantized via Zeta Criticality in the sense of the Riemann Hypothesis. I assert that if the Riemann Hypothesis is true, then the following must be equivalent:

- The discontinuity of the Potential of a Riemann Magneton at the Critical Value ½;
- The deflection coefficient of a photon around a rotating black hole with a fixed scalar Lyapunov exponent.

As a Corollary: If the Riemann Hypothesis is true, then there is a strong correspondence between Riemann Magnetons (theoretical mathematics) and all possible uncharged, rotating black holes of Euclidean Dimension 4.

This result is remarkable because the Riemann Magneton is explicitly determined by Faraday’s Laws and is a result of electromagnetic forces. The Kerr Black Hole is strictly a function of Einstein’s Field Equations with a rotation parameter. Such serendipity might suggest that the results for the Photon Deflection around charged black holes might be resolvable in a purely electromagnetic paradigm parametrized by the Riemann Zeta Function. If you can find a mistake, please write me straight away. Perhaps some think-tanks in the Valley can generalize it or run numerical experiments on a massive scale that I do not have the resources to implement.

Technical Statement: Suppose that the Riemann Hypothesis is true. Then the following must be equivalent:

- The Merlini Jump (2004) of a Riemann Magneton at critical value ½
- The Lyapunov Exponent of Sneppen (2021) for a Kerr Black Hole on an Einstein Manifold.
- Moreover, every Normal Distribution of Standard Variance S=(4*pi) corresponds to a Lyapunov exponent L which is of the form L^(1/2) = 2 + y – ln|S|, where y is the Euler-Mascheroni Constant. This constant is the exponent for the dominant Eigenvalue of Sneppen’s Divergent-Reflection paradigm in the Kerr Case.

SDG

[1]. Misner, C., Thorne, K. and Wheeler, J. (2017) Gravitation. Princeton University Press, Princeton.

[2]. Merlini (Preprint, 2004): https://arxiv.org/abs/math-ph/0404031

[2]. Sneppen, A. “Divergent reflections around the photon sphere of a black hole”. *Sci Rep ***11**, 14247 (2021). https://doi.org/10.1038/s41598-021-93595-w

[3]. Sondow, Jonathan (1998)”An antisymmetric formula for Euler’s constant”. *Mathematics Magazine*.**71**(3): 219–220.doi:10.1080/0025570X.1998.11996638

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