Mathematics Letters

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The Geometric Interpretation of Some Mathematical Expressions Containing the Riemann ζ-Function

Received: 06 October 2016    Accepted: 10 November 2016    Published: 17 December 2016
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Abstract

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.

DOI 10.11648/j.ml.20160206.11
Published in Mathematics Letters (Volume 2, Issue 6, December 2016)
Page(s) 42-46
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Riemann ζ-Function, Einstein Equations, Metric, Metric Tensor, Energy-Momentum Tensor, Christoffel Symbols, Algorithm, Turing Machine

References
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[2] Janke E., Emde F, Lösch F., Tafeln Höherer Funktionen, B.G. Teubner Verlagsgeselschaft, Stuttgart, 1960.
[3] Hardy G. H. Divergent series.-Oxford, 1949.
[4] Eiler L. Differential Calculation.- Academy Pub., St. Petersburg.- 1755.
[5] Landau L. D., Lifshitz E. M., The Classical Theory of Fields. Vol. 2 (4th ed.). Butterworth-Heinemann, 1975.
[6] Maxwell’s Demon 2. Entropy, Classical and Quantum Information, Computing. Ed. by Leff H.S., and Rex A.F., IoP Publishing, 2003.
[7] Prudnikov A. P., Brychkov Yu. A., and Marichev O. I., Integrals and Series, vols. 1–3; Gordon and Breach, New York, 1986, 1986, 1989.
[8] Derbyshire J., Prime obsession. Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Joseph Henry Press, Washington, D.C., 2003.
[9] ZetaGrid Homepage: http://www.wstein.org/simuw/misc/ zeta_grid.html
[10] 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.
[11] 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.
[12] 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.
[13] 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.
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Author Information
  • Department of Applied Informatics, Faculty of Public Administration, The Russian Presidential Academy of National Economy and Public Administration, Saratov Branch, Saratov, Russia

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  • APA Style

    Yuriy N. Zayko. (2016). The Geometric Interpretation of Some Mathematical Expressions Containing the Riemann ζ-Function. Mathematics Letters, 2(6), 42-46. https://doi.org/10.11648/j.ml.20160206.11

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    ACS Style

    Yuriy N. Zayko. The Geometric Interpretation of Some Mathematical Expressions Containing the Riemann ζ-Function. Math. Lett. 2016, 2(6), 42-46. doi: 10.11648/j.ml.20160206.11

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    AMA Style

    Yuriy N. Zayko. The Geometric Interpretation of Some Mathematical Expressions Containing the Riemann ζ-Function. Math Lett. 2016;2(6):42-46. doi: 10.11648/j.ml.20160206.11

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  • @article{10.11648/j.ml.20160206.11,
      author = {Yuriy N. Zayko},
      title = {The Geometric Interpretation of Some Mathematical Expressions Containing the Riemann ζ-Function},
      journal = {Mathematics Letters},
      volume = {2},
      number = {6},
      pages = {42-46},
      doi = {10.11648/j.ml.20160206.11},
      url = {https://doi.org/10.11648/j.ml.20160206.11},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ml.20160206.11},
      abstract = {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.},
     year = {2016}
    }
    

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