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Aperiodic crystals, Riemann zeta function, and primes

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Abstract

The distribution of Riemann zeros may represent a spectrum of a certain point distribution. Potentially suitable distributions were synthesized by summing complex exponentials over Riemann zeros. If the magnitudes of all harmonics are equal to each other, the resulting spectrum has peaks at the logarithms of primes and prime powers in accordance with the von Mangoldt function. If the magnitudes are set inversely proportional to the derivative of the Riemann zeta function at zeros, the spectrum has peaks at the logarithms of primes and products of distinct primes following the Möbius function. Combining trigonometric series over Riemann zeros with the Möbius function, we obtained the spectrum that consists of equal intensity peaks at the logarithms of primes only.

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Acknowledgements

Authors thank Jelena R. Kambak for proofreading and writing assistance.

Funding

This work was partially supported by the Ministry of Science and Higher Education of the Russian Federation (0784-2020-0025 to S.V.K., A.E.M.) and the Foundation for Assistance to Small Innovative Enterprises in Science and Technology, Program UMNIK (0062155 to P.A.M.).

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Authors and Affiliations

  1. Peter the Great St. Petersburg Polytechnic University, ul. Polytechnicheskaya 29, Saint Petersburg, 195251, Russia

    Alexey E. Madison & Sergey V. Kozyrev

  2. Saint Petersburg Electrotechnical University “LETI”, ul. Professora Popova 5, Saint Petersburg, 197376, Russia

    Pavel A. Madison

Authors
  1. Alexey E. Madison
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  2. Pavel A. Madison
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Contributions

A. E. M. carried out conceptualization, developed methodology, took the leading role in formal analysis and investigation; P. A. M. carried out computer programming, prepared computer graphics; S. V. K. supervised the work, participated in conceptualization. The original draft of the manuscript was written by A. E. M., and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Alexey E. Madison.

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Madison, A.E., Madison, P.A. & Kozyrev, S.V. Aperiodic crystals, Riemann zeta function, and primes. Struct Chem 34, 777–790 (2023). https://doi.org/10.1007/s11224-022-01906-2

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  • DOI: https://doi.org/10.1007/s11224-022-01906-2

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