Number Theory Number Symmetry and • Michel Planat Number Theory and Symmetry Edited by Michel Planat Printed Edition of the Special Issue Published in Symmetry www.mdpi.com/journal/symmetry Number Theory and Symmetry Number Theory and Symmetry Editor Michel Planat MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editor Michel Planat Institut FEMTO-ST France Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Symmetry (ISSN 2073-8994) (available at: https://www.mdpi.com/journal/symmetry/special issues/Number Theory Symmetry). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03936-686-6 (Hbk) ISBN 978-3-03936-687-3 (PDF) c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Editor .............................................. vii Preface to ”Number Theory and Symmetry” .............................. ix Germ´anSierra The Riemann Zeros as Spectrum and the Riemann Hypothesis Reprinted from: Symmetry 2019, 11, 494, doi:10.3390/sym11040494 ................. 1 Michel Planat, Raymond Aschheim, Marcelo M. Amaral and Klee Irwin Universal Quantum Computing and Three-Manifolds Reprinted from: Symmetry 2018, 10, 773, doi:10.3390/sym10120773 ................. 39 Torsten Asselmeyer-Maluga Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics Reprinted from: Symmetry 2019, 11, 1298, doi:10.3390/sym11101298 ................ 55 Bruno Aiazzi, Stefano Baronti, Leonardo Santurri and Massimo Selva An Investigation on the Prime and Twin Prime Number Functions by Periodical Binary Sequences and Symmetrical Runs in a Modified Sieve Procedure Reprinted from: Symmetry 2019, 11, 775, doi:10.3390/sym11060775 ................. 81 S.T. Ishmukhametov, B.G. Mubarakov and R.G. Rubtsova On the Number of Witnesses in the Miller–Rabin PrimalityTest Reprinted from: Symmetry 2020, 12, 890, doi:10.3390/sym12060890 .................103 Atsushi Yamagami and Kazuki Taniguchi On a Generalization of a Lucas’ Resultand an Application to the 4-Pascal’s Triangle Reprinted from: Symmetry 2020, 12, 288, doi:10.3390/sym12020288 .................115 Pavel Trojovsky´ Algebraic Numbers as Product of Powers of Transcendental Numbers Reprinted from: Symmetry 2019, 11, 887, doi:10.3390/sym11070887 .................123 Atsushi Yamagami and Y ¯ukiMatsui On Some Formulas for Kaprekar Constants Reprinted from: Symmetry 2019, 11, 885, doi:10.3390/sym11070885 .................129 Ilwoo Cho ∗ Asymptotic Semicircular Laws Induced by p-Adic Number Fields Qpand C -Algebras over Primes p Reprinted from: Symmetry 2019, 11, 819, doi:10.3390/sym11060819 .................161 v About the Editor Michel Planat was a researcher at the National Center of Scientific Research in France from 1982 to 2018. From 1980 to 2001, he researched nonlinear waves in piezoelectric crystals and 1/f noise in quartz resonators. He established links between 1/f noise and number theory. He also researched the Riemann hypothesis. He discovered Ramanujan sums signal processing. From 2002 to 2018, he was interested in quantum information theory with work concerning mutually unbiased bases, quantum entanglement and contextuality, and quantum computing, using mathematical tools such as finite geometries, number theory, dessin d’enfants, and free group theory. Since 2019, he is a visiting scientist at FEMTO-ST Institute in Besanc¸on, France. He is also an associate research scientist at Quantum Gravity Research in Los Angeles. His current research is about topological quantum computing from three- and four-manifolds to establish bridges between quantum computing and quantum gravity. vii Preface to ”Number Theory and Symmetry” “Number Theory and Symmetry” deals with topics connecting numbers (integers, algebraic integers, transcendental numbers, p-adic numbers) and symmetries. First of all, symmetry became part of number theory when Riemann investigated the distribution of prime numbers and for that purpose introduced the complex functional equation and the related Riemann hypothesis (RH) that non-trivial zeros of the Riemann zeta function lie on the symmetry axis s = 1/2. Then, in a quest to justify RH on physical grounds, the Hilbert–Polya conjecture claimed that the imaginary part of the Riemann zeros on the symmetry axis should correspond to the eigenvalues of a Hermitian operator. This topic is covered by German Sierra. Besides these classical areas, number fields offer clues to the connection between numbers and symmetries through arithmetic groups, geometry, and topology. I have in mind the Poincare´ conjecture and the whole work of Thurston about 3-manifolds. This topic is the kernel of the two papers by Michel Planat and co-authors and Torsten Asselmeyer Maluga. The aforementioned three papers highlight a strong connection between number theory and quantum physics. The range of the three subsequent papers in this series is about more standard topics of number theory. A modified Sieve procedure by Bruno Aiazzi and coauthors, the Miller–Rabin primality test by Shamil Ishmukhametov and co-authors, and the 4-Pascal’s triangle by Atsushi Yamagami and Kazuki Taniguchi are investigated. The paper by Pavel Trojovsky offers clues to the relation between algebraic and transcendental numbers through polynomials. Atsushi Yamagami and Yuki Matsui’s paper is in the field of b-adic numbers. The last paper by Ilwoo Cho covers the topic of p-adic numbers thanks to C*-algebras and Banach*-probability spaces. The rich panel of mathematical concepts involved in this Special Issue illustrates the continuous interest of scholars in the relationship between numbers, their symmetries, and physics. Michel Planat Editor ix S S symmetry Article The Riemann Zeros as Spectrum and the Riemann Hypothesis Germán Sierra Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain; [email protected] Received: 31 December 2018; Accepted: 26 March 2019; Published: 4 April 2019 Abstract: We present a spectral realization of the Riemann zeros based on the propagation of a massless Dirac fermion in a region of Rindler spacetime and under the action of delta function potentials localized on the square free integers. The corresponding Hamiltonian admits a self-adjoint extension that is tuned to the phase of the zeta function, on the critical line, in order to obtain the Riemann zeros as bound states. The model suggests a proof of the Riemann hypothesis in the limit where the potentials vanish. Finally, we propose an interferometer that may yield an experimental observation of the Riemann zeros. Keywords: zeta function; Pólya-Hilbert conjecture; Riemann interferometer 1. Introduction One of the most promising approaches to prove the Riemann Hypothesis [1–7] is based on the conjecture, due to Pólya and Hilbert, that the Riemann zeros are the eigenvalues of a quantum mechanical Hamiltonian [8]. This bold idea is supported by several results and analogies involving Number Theory, Random Matrix Theory and Quantum Chaos [9–17]. However, the construction of a Hamiltonian whose spectrum contains the Riemann zeros, has eluded researchers for several decades. In this paper we shall review the progress made along this direction starting from the famous xp model proposed in 1999 by Berry, Keating and Connes [18–20] that inspired many works [21–45], some of them will be discuss below. See [46] for a general review on physical approaches to the RH. Other approaches to the RH and related material can be found in [47–63]. To relate xp with the Riemann zeros, Berry, Keating and Connes used two different regularizations. The Berry and Keating regularization led to a discrete spectrum related to the smooth Riemann zeros [18,19], while Connes’s regularization led to an absorption spectrum where the zeros are missing spectral lines [20]. A physical realization of the Connes model was obtained in 2008 in terms of the dynamics of an electron moving in two dimensions under the action of a uniform perpendicular magnetic field and an electrostatic potential [29]. However this model has not been able to reproduce the exact location of the Riemann zeros. On the other hand, the Berry–Keating xp model was revisited in 2011 in terms of the classical Hamiltonians H = x(p + 1/p), and H =(x + 1/x)(p + 1/p) whose quantizations contain the smooth approximation of the Riemann zeros [32,36]. Later on, these models were generalized in terms of the family of Hamiltonians H = U(x)p + V(x)/p that were shown to describe the dynamics of a massive particle in a relativistic spacetime whose metric can be constructed using the functions U and V [35]. This result suggested a reformulation of H = U(x)p + V(x)/p in terms of the massive Dirac equation in the aforementioned spacetimes [38]. Using this reformulation, the Hamiltonian H = x(p + 1/p) was shown to be equivalent to the massive Dirac equation in Rindler spacetime that is the natural arena to study accelerated observers and the Unruh effect [42]. This result provides an appealing spacetime interpretation of the xp model and in particular of the smooth Riemann zeros. Symmetry 2019, 11, 494; doi:10.3390/sym110404941 www.mdpi.com/journal/symmetry Symmetry 2019, 11, 494 To obtain the exact zeros, one must make further modifications of the Dirac model. First, the fermion must become massless. This change is suggested by a field theory interpretation of the Pólya’s ξ function and its comparison with the Riemann’s ξ function.