Abstract
In this article, the proof of the Riemann hypothesis is considered using the calculation of the Riemann ζ-function on a relativistic computer. 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. Considerations are given in favor of the validity of the Riemann hypothesis with respect to the distribution of non-trivial zeros of the ζ-function.
Highlights
In the article [1] a relativistic Turing machine was suggested for calculation the Riemann ζ-function presented by the series which diverges for u < 1 [2]. ∑ ∞ζ (w) = n−w, w = u + iv n=1Such an approach to solving problems beyond the range of problems solved by the classical Turing machine develops in a number of works since the 1980s of the last century
One can imagine a relativistic Turing machine, the role of the head of which is performed by a material particle moving in accordance with relativistic equations of motion determined by the corresponding physical problem
Like in the previous paper of the author [1] the computation of the Riemann ζ-function represented by a divergent series in the plane of the complex argument w=u + iv is performed using the methods of the general theory of relativity
Summary
In the article [1] a relativistic Turing machine was suggested for calculation the Riemann ζ-function presented by the series which diverges for u < 1 [2]. Such an approach (known as "Beyond Turing") to solving problems beyond the range of problems solved by the classical Turing machine develops in a number of works since the 1980s of the last century (see, for example, [3]). One can imagine a relativistic Turing machine, the role of the head of which is performed by a material particle moving in accordance with relativistic equations of motion determined by the corresponding physical problem. Considerations are given in favor of the validity of the Riemann hypothesis with respect to the distribution of nontrivial zeros of the ζ-function
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