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On Some Algebraic Ways to Calculate Zeros of the Riemann Zeta Function

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Algebraic Informatics (CAI 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13706))

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Abstract

The Riemann zeta function is an important number-theoretical tool for studying prime numbers. The first part of the paper is a short survey of some known results about this function. The emphasis is given to the possibility to formulate the celebrated Riemann Hypothesis as a statement from class \(\mathrm {\Pi }^0_1\) in the arithmetical hierarchy.

In the second part of the paper the author demonstrates by numerical examples some non-evident ways for finding zeros of the zeta function. Calculations require the knowledge of the value of this function and of N its initial derivatives at one point and consist in solving N systems of linear equations with N unknowns.

These methods are not intended for practical calculations but are supposed to be useful for the study of the zeros.

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Notes

  1. 1.

    After M. Davis, H. Putnam, J. Robinson and Yu. Matiyasevich; for detailed proofs see, for example, [14, 19, 23, 25, 30].

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Authors and Affiliations

  1. Petersburg Department of Steklov Institute of Mathematics, St. Petersburg, 191023, Russia

    Yuri Matiyasevich

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  1. Yuri Matiyasevich
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Correspondence to Yuri Matiyasevich .

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  1. Aristotle University of Thessaloniki, Thessaloniki, Greece

    Dimitrios Poulakis

  2. Aristotle University of Thessaloniki, Thessaloniki, Greece

    George Rahonis

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Matiyasevich, Y. (2022). On Some Algebraic Ways to Calculate Zeros of the Riemann Zeta Function. In: Poulakis, D., Rahonis, G. (eds) Algebraic Informatics. CAI 2022. Lecture Notes in Computer Science, vol 13706. Springer, Cham. https://doi.org/10.1007/978-3-031-19685-0_2

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  • DOI: https://doi.org/10.1007/978-3-031-19685-0_2

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