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 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 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.
Highlights
In this paper, an attempt is made to calculate the Riemann ζ-function using relativistic effects, or, more precisely, effects of the general theory of relativity (GRT)
In this article the computation of the Riemann ζ-function represented by a divergent series in the plane of the complex argument is performed using the methods of the general theory of relativity
The calculation is realized by the motion of some material particle in a curved metric
Summary
An attempt is made to calculate the Riemann ζ-function using relativistic effects, or, more precisely, effects of the general theory of relativity (GRT). This idea was first used in the author's paper [1], where it was shown that by introducing the corresponding curved metric at the axis of real numbers, one can calculate the sum of a divergent series representing ζ (-1). The calculation of the sum of a divergent series (a non-computable problem) is regarded as a physical problem about the motion of a material point in curved space-time, which is, the embodiment of the thesis, which is inverse to the well-known thesis, that any motion of a physical system can be treated as calculation [4]. Zayko: Calculation of the Riemann Zeta-function on a Relativistic Computer
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