The Geometric Interpretation of Some Mathematical Expressions Containing the Riemann ζ-Function
Volume 2, Issue 6, December 2016, Pages: 42-46
Received: Oct. 6, 2016;
Accepted: Nov. 10, 2016;
Published: Dec. 17, 2016
Views 3129 Downloads 98
Yuriy N. Zayko, Department of Applied Informatics, Faculty of Public Administration, The Russian Presidential Academy of National Economy and Public Administration, Saratov Branch, Saratov, Russia
Follow on us
The article discusses some of the mathematical results widely used in practice which contain the Riemann ζ-function, and, at first glance, are in contradiction with common sense. A geometric approach is suggested, based on the concept of the curvature of space, in which is calculated an algorithm that specifies the representation of ζ-function as an infinite diverging series. The analysis is based on the use of Einstein equations to calculate the metric of curved space-time. The solution of the Einstein equations is a metric that has a singularity, like the metric in the vicinity of the black hole. The result can be interpreted in the spirit of a Turing machine that performs the proposed algorithm for calculating the sum of a divergent series.
Riemann ζ-Function, Einstein Equations, Metric, Metric Tensor, Energy-Momentum Tensor, Christoffel Symbols, Algorithm, Turing Machine
To cite this article
Yuriy N. Zayko,
The Geometric Interpretation of Some Mathematical Expressions Containing the Riemann ζ-Function, Mathematics Letters.
Vol. 2, No. 6,
2016, pp. 42-46.
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Zwiebach B. A First Course in String Theory, 2-nd Ed., MIT.- 2009.
Janke E., Emde F, Lösch F., Tafeln Höherer Funktionen, B.G. Teubner Verlagsgeselschaft, Stuttgart, 1960.
Hardy G. H. Divergent series.-Oxford, 1949.
Eiler L. Differential Calculation.- Academy Pub., St. Petersburg.- 1755.
Landau L. D., Lifshitz E. M., The Classical Theory of Fields. Vol. 2 (4th ed.). Butterworth-Heinemann, 1975.
Maxwell’s Demon 2. Entropy, Classical and Quantum Information, Computing. Ed. by Leff H.S., and Rex A.F., IoP Publishing, 2003.
Prudnikov A. P., Brychkov Yu. A., and Marichev O. I., Integrals and Series, vols. 1–3; Gordon and Breach, New York, 1986, 1986, 1989.
Derbyshire J., Prime obsession. Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Joseph Henry Press, Washington, D.C., 2003.
ZetaGrid Homepage: http://www.wstein.org/simuw/misc/ zeta_grid.html
Pei-Chu Hu and Bao Qin Li. A Connection between the Riemann Hypothesis and Uniqueness of the Riemann zeta function, arXiv:1610.01583v1 [math.NT] 5 Oct 2016.
McPhedran R.C., Zeros of Lattice Sums: 3. Reduction of the Generalised Riemann Hypothesis to Specific Geometries, arXiv:1610.07932v1 [math-ph] 22 Oct 2016.
May M. P., On the Location of the Non-Trivial Zeros of the RH via Extended Analytic Continuation, arXiv:1608.08082v3 [math.GM] 16 Sep 2016.
Andr´eka H., N´emeti I., N´emeti P. General relativistic hypercomputing and foundation of mathematics, Natural Computing, 2009, V. 8, № 3, pp 499–516.
Weinberg S., Gravitation and Cosmology. Principles and Applications of the General Theory of Relativity, MTI, John Villey & Sons, NY, 1972.