Riemann Hypothesis from the Physicist’s Point of View
Keywords:Riemann zeta-function, Riemann hypothesis, non-trivial zeroes, Turing barrier, relativistic computations, horizon
This article presents an attempt to comprehend the evolution of the ideas underlying the physical approach to the proof of one of the problems of the century - the Riemann hypothesis regarding the location of non-trivial zeros of the Riemann zeta function. Various formulations of this hypothesis are presented, which make it possible to clarify its connection with the distribution of primes in the set of natural numbers. A brief overview of the main directions of this approach is given. The probable cause of their failures is indicated - the solution of the problem within the framework of the classical Turing paradigm. A successful proof of the Riemann hypothesis based on the use of a relativistic computation model that allows one to overcome the Turing barrier is presented. This model has been previously applied to solve another problem not computable on the classical Turing machine - the calculation of the sums of divergent series for the Riemann zeta function of the real argument. The possibility of using relativistic computing for the development of artificial intelligence systems is noted.
(1) Janke E., Emde F., Lösch F., Tafeln Höherer Funktionen, B.G. Teubner Verlagsgeselschaft, Stuttgart, 1960.
(2) Riemann Bernhard, Ueber die Anzahl der Primzahlen unter einer gegebenen Gr¨osse, Monatsberichte der Berliner Akademie, Nov. 1859. URL: http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/
(3) Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Joseph Henry Press, Washington, DC, 2003.
(4) Jinhua Fei, Riemann hypothesis is not correct, arXiv:1407. 4545v1 [math. GM] 17 Jul 2014
(5) Pr´astaro, A., The Riemann Hypothesis Proved, arXiv:1305. 6845v10 [math. GM] 27 Oct 2015.
(6) McPhedran, R. C., Constructing a Proof of the Riemann Hypothesis, arXiv:1309. 5845v1 [math. NT] 30 Aug 2013.
(7) Lee, Jin Gyu, The Riemann Hypothesis and the possible proof, arXiv:1402. 2822v1 [math. GM] 9 Feb 2014.
(8) Blinovsky, V., Proof of Riemann hypothesis, arXiv:1703. 03827v5 [math. GM] 1 May 2017.
(9) Stenger, F., A Proof of the Riemann Hypothesis, arXiv:1708. 01209v2 [math. GM] 14 Aug 2017.
(10) Wolf, M., Will a physicist prove the Riemann Hypothesis?, arXiv:1410. 1214v3 [math-ph] 1 Dec 2015.
(11) Mackay, R. S., Towards a Spectral Proof of Riemann's Hypothesis, arXiv:1708. 00440v1 [math. SP] 1 Aug 2017.
(12) Matiyasevich, Yu. V., Alan Turing and Number Theory (to the 100 anniversary of A. Turing’s birth), The Alan Turing Centenary Conference (Manchester, UK, June 22–25 2012); Published in: Mathematics in Higher Education,
, № 10, 111-134 (Russian).
(13) Y.N. Zayko. The Proof of the Riemann Hypothesis on a Relativistic Turing Machine. International Journal of Theoretical and Applied Mathematics, 2017, vol 3, № 6, pp. 219-224.
(14) Zayko, Y.N., Capabilities of a Relativistic Computer. 18th International Conference named after A.F. Terpugov “Information Technologies and Mathematical Modelling”( ITMM-2019 ) Saratov, 2019, June 26-30. Tomsk: NTL Publishing, 2019. Part. 1, pp.175-180
(15) Sabbagh, K., The Riemann Hypothesis: the Greatest Unsolved Problem in Mathematics. Farrar, Strauss and Giroux, New York, 2003, 342 p.
(16) Mazur, B. and W. Stein. Prime Numbers and the Riemann Hypothesis. Cambridge Univ. Press, New York, NY, 2016, 142 p.
(17) Sierra, G., A Physics Pathway to the Riemann Hypothesis, Fisica Teorica, Julio Abad, 2008. pp. 1 – 8; arXiv: 1012.4264v1 [math-ph] 20 Dec 2010
(18) Keating, J. P., Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Physics Today, 2004, 57, 6, 63; doi: 10.1063/1.1784281
(19) Berry, M. V., Riemann’s zeta function: a model for quantum chaos? Quantum Chaos and Statistical Nuclear Physics ed. T H Seligman and H Nishioka, 1986, vol. 263, pp. 1-17.
(20) Berry, M.V. Semiclassical formula for the number variance of the Riemann zeros. Nonlinearity, 1988,v. 1, pp. 399-407.
(21) Berry, M. V. Keating J., H = xp and the Riemann zeros in Supersymmetry and trace formulae: chaos and disorder, I. V. Lerner and J. P. Keating, eds., New York: Plenum, 1999.
(22) Atiyah, M., URL: https://drive.google.com/file/d/17NBICP6OcUSucrXKNW vzLmr QpfUrEKuY/view
(23) Zayko, Y. N., Calculation of the Riemann Zeta-function on a Relativistic Computer. Mathematics and Computer Science,2017, V. 2, № 2, pp. 20-26.
(24) Landau, L. D. and E. M. Lifshitz. The Classical Theory of Fields, (4th ed.). Butterworth-Heinemann, 1975, 458 p.
(25) Zayko, Y.N., Maxwell’s Electrodynamics in Curved Space-Time. World Journal of Innovative Research, 2016, V. 1, № 1, pp. 16-19.
(26) Nemeti, I. and G. David. Relativistic Computers and the Turing Barrier. Applied Mathematics and Computation. 2006, V.178, pp. 118-142.
(27) Zayko, Y.N., The Geometric Interpretation of Some Mathematical Expressions Containing the Riemann ζ-Function. Mathematics Letters, 2016, V. 2, № 6, pp. 42-46.
(28) Zayko, Y.N., Quantum Field Theory Free of Divergences. Global Journal of Science Frontier Research: A - Physics & Space Science, 2020, V. 20, № 1 (Ver. 1.0), pp. 27 – 31.
(29) Penrose, R., Shadows of the Mind. A Search for the Missing Science of Consciousness. Oxford Univ. Press, NY, Oxford, 1989.
(30)Penrose, R., The Emperor’s New Mind. Concerning Computers, Minds and The Laws of Physics. Oxford Univ. Press, NY, Oxford, 1989