Calculation of the Riemann Zeta-function on a Relativistic Computer

Yuriy N. Zayko

doi:doi:10.11648/j.mcs.20170202.12

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Calculation of the Riemann Zeta-function on a Relativistic Computer

Yuriy N. Zayko

Published in Mathematics and Computer Science (Volume 2, Issue 2)

Received: 10 May 2017 Accepted: 20 May 2017 Published: 6 July 2017

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Abstract

The problem of calculating the sum of a divergent series for the Riemann ζ-function of a complex argument is considered in the paper, using the effects of the general theory of relativity. The parameters of the reference frame metric in which the calculation is performed are determined and solutions of the relativistic equations of motion of the material point realizing the calculation are found. The work lies at the junction of the direction known as "Beyond Turing", considering the application of the so-called "relativistic supercomputers" for solving non-computable problems and a direction devoted to the study of non-trivial zeros of the Riemann ζ-function. The formulation of the Riemann hypothesis concerning the distribution of nontrivial zeros of the ζ-function from the point of view of their computability on a relativistic computer is given. In view of the importance of the latter issue for studying the distribution of prime numbers, the results of the work may be of interest to specialists in the field of information security.

Published in	Mathematics and Computer Science (Volume 2, Issue 2)
DOI	10.11648/j.mcs.20170202.12
Page(s)	20-26
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), 2017. Published by Science Publishing Group

Keywords

Metric, Riemann zeta-function, General Theory of Relativity, Space-Time Curvature, Non-computable Problems, Singularity, Black Hole, Relativistic Computer

References

[1]	Y. N. Zayko, The Geometric Interpretation of Some Mathematical Expressions Containing the Riemann ζ-Function. Mathematics Letters, 2016; 2(6): 42-46.
[2]	G. H. Hardy, Divergent series, Oxford, 1949.
[3]	I. Nemeti, G. David, Relativistic computers and the Turing barrier. Applied Mathematics and Computation, 178, 118-142, 2006.
[4]	S. Lloyd, Programming the Universe. A Quantum Computer Scientist Takes on the Cosmos, Knopf, 2006.
[5]	W. J. Cody, K. E. Hillstrom, Thacher, C. Henry, Jr, Chebyshev approximations for the Riemann zeta function. Math. Comp. -1971.-№25. С. 537–547.
[6]	R. L. Graham, D. E. Knuth, O. Patashnick, Concrete Mathematics. A Foundation for Computer Science, 2^nd Ed, Addison-Wesley, 1994.
[7]	E. A. Karatsuba, Rapid computation of the Riemann zeta-function ζ (s) for integer values of the argument s, Problems of information transfer. -1995. -№31.-P. 69-80. (Russian).
[8]	A. A. Karatsuba, Foundations of Analytic Number Theory. – Moscow, Science, 1983. (Russian).
[9]	E. Janke, F. Emde, F. Lösch, Tafeln Höherer Funktionen, B. G. Teubner Verlagsgeselschaft, Stuttgart, 1960.
[10]	L. G. Loitsyansky, Mechanics of Liquids and Gases, Moscow, Leningrad, GITTL, 1950 (Russian).
[11]	L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields, (4th ed.), Butterworth-Heinemann, 1975.
[12]	R. C. Tolman, Relativity Thermodynamics and Cosmology, Clarendon Press, Oxford, 1969.
[13]	John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, J. H. Press, 2003.

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APA Style

Yuriy N. Zayko. (2017). Calculation of the Riemann Zeta-function on a Relativistic Computer. Mathematics and Computer Science, 2(2), 20-26. https://doi.org/10.11648/j.mcs.20170202.12

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ACS Style

Yuriy N. Zayko. Calculation of the Riemann Zeta-function on a Relativistic Computer. Math. Comput. Sci. 2017, 2(2), 20-26. doi: 10.11648/j.mcs.20170202.12

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AMA Style

Yuriy N. Zayko. Calculation of the Riemann Zeta-function on a Relativistic Computer. Math Comput Sci. 2017;2(2):20-26. doi: 10.11648/j.mcs.20170202.12

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@article{10.11648/j.mcs.20170202.12,
  author = {Yuriy N. Zayko},
  title = {Calculation of the Riemann Zeta-function on a Relativistic Computer},
  journal = {Mathematics and Computer Science},
  volume = {2},
  number = {2},
  pages = {20-26},
  doi = {10.11648/j.mcs.20170202.12},
  url = {https://doi.org/10.11648/j.mcs.20170202.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20170202.12},
  abstract = {The problem of calculating the sum of a divergent series for the Riemann ζ-function of a complex argument is considered in the paper, using the effects of the general theory of relativity. The parameters of the reference frame metric in which the calculation is performed are determined and solutions of the relativistic equations of motion of the material point realizing the calculation are found. The work lies at the junction of the direction known as "Beyond Turing", considering the application of the so-called "relativistic supercomputers" for solving non-computable problems and a direction devoted to the study of non-trivial zeros of the Riemann ζ-function. The formulation of the Riemann hypothesis concerning the distribution of nontrivial zeros of the ζ-function from the point of view of their computability on a relativistic computer is given. In view of the importance of the latter issue for studying the distribution of prime numbers, the results of the work may be of interest to specialists in the field of information security.},
 year = {2017}
}

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TY  - JOUR
T1  - Calculation of the Riemann Zeta-function on a Relativistic Computer
AU  - Yuriy N. Zayko
Y1  - 2017/07/06
PY  - 2017
N1  - https://doi.org/10.11648/j.mcs.20170202.12
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Author Information

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

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APA Style

Yuriy N. Zayko. (2017). Calculation of the Riemann Zeta-function on a Relativistic Computer. Mathematics and Computer Science, 2(2), 20-26. https://doi.org/10.11648/j.mcs.20170202.12

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ACS Style

Yuriy N. Zayko. Calculation of the Riemann Zeta-function on a Relativistic Computer. Math. Comput. Sci. 2017, 2(2), 20-26. doi: 10.11648/j.mcs.20170202.12

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AMA Style

Yuriy N. Zayko. Calculation of the Riemann Zeta-function on a Relativistic Computer. Math Comput Sci. 2017;2(2):20-26. doi: 10.11648/j.mcs.20170202.12

Copy | Download

@article{10.11648/j.mcs.20170202.12,
  author = {Yuriy N. Zayko},
  title = {Calculation of the Riemann Zeta-function on a Relativistic Computer},
  journal = {Mathematics and Computer Science},
  volume = {2},
  number = {2},
  pages = {20-26},
  doi = {10.11648/j.mcs.20170202.12},
  url = {https://doi.org/10.11648/j.mcs.20170202.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20170202.12},
  abstract = {The problem of calculating the sum of a divergent series for the Riemann ζ-function of a complex argument is considered in the paper, using the effects of the general theory of relativity. The parameters of the reference frame metric in which the calculation is performed are determined and solutions of the relativistic equations of motion of the material point realizing the calculation are found. The work lies at the junction of the direction known as "Beyond Turing", considering the application of the so-called "relativistic supercomputers" for solving non-computable problems and a direction devoted to the study of non-trivial zeros of the Riemann ζ-function. The formulation of the Riemann hypothesis concerning the distribution of nontrivial zeros of the ζ-function from the point of view of their computability on a relativistic computer is given. In view of the importance of the latter issue for studying the distribution of prime numbers, the results of the work may be of interest to specialists in the field of information security.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Calculation of the Riemann Zeta-function on a Relativistic Computer
AU  - Yuriy N. Zayko
Y1  - 2017/07/06
PY  - 2017
N1  - https://doi.org/10.11648/j.mcs.20170202.12
DO  - 10.11648/j.mcs.20170202.12
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 20
EP  - 26
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20170202.12
AB  - The problem of calculating the sum of a divergent series for the Riemann ζ-function of a complex argument is considered in the paper, using the effects of the general theory of relativity. The parameters of the reference frame metric in which the calculation is performed are determined and solutions of the relativistic equations of motion of the material point realizing the calculation are found. The work lies at the junction of the direction known as "Beyond Turing", considering the application of the so-called "relativistic supercomputers" for solving non-computable problems and a direction devoted to the study of non-trivial zeros of the Riemann ζ-function. The formulation of the Riemann hypothesis concerning the distribution of nontrivial zeros of the ζ-function from the point of view of their computability on a relativistic computer is given. In view of the importance of the latter issue for studying the distribution of prime numbers, the results of the work may be of interest to specialists in the field of information security.
VL  - 2
IS  - 2
ER  -

Copy | Download