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On Hyperbolic Zeta Function of Lattices

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Continuous and Distributed Systems

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 211))

Abstract

This chapter provides an overview of the theory of hyperbolic zeta function of lattices. A functional equation for the hyperbolic zeta function of Cartesian lattice is obtained. Information about the history of the theory of the hyperbolic zeta function of lattices is provided. The relations with the hyperbolic zeta function of nets and Korobov optimal coefficients are considered.

Dedicated to the 95th Birth Anniversary

of Nikolai Mikhailovich Korobov

\(\mathrm{(23.11.1917}\)–\(\mathrm{25.10.2004)}\)

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Notes

  1. 1.

    Here and hereafter \(\sum '\) denotes summation over systems: \((m_1,\ldots ,m_s)\ne (0,\ldots ,0).\)

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Acknowledgments

The authors are grateful to professor G. I. Arkhipov and to professorV. N. Chubarikov for constant attention to this work and for useful discussions. This research was partially supported by the RFBR grant 11-01-00571.

Author information

Authors and Affiliations

  1. Institute of Economics and Management, 10, Veresaeva St., Tula, Russia, 300041

    L. P. Dobrovolskaya

  2. Geophysical center of RAS, 3, Molodezhnaya St., Moscow, Russia, 119296

    M. N. Dobrovolsky

  3. Leo Tolstoy Tula State Pedagogical University, 125, Lenina pr., Tula, Russia, 300026

    N. M. Dobrovol’skii

  4. Tula State University, 92, Lenina pr., Tula, Russia, 300012

    N. N. Dobrovolsky

Authors
  1. L. P. Dobrovolskaya
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  2. M. N. Dobrovolsky
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  3. N. M. Dobrovol’skii
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  4. N. N. Dobrovolsky
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Correspondence to M. N. Dobrovolsky .

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Editors and Affiliations

  1. National Academy of Science, National Technical University of Ukraine, Kiev, Ukraine

    Mikhail Z. Zgurovsky

  2. Lomonosov Moscow State University, Moscow, Russia

    Victor A. Sadovnichiy

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Dobrovolskaya, L.P., Dobrovolsky, M.N., Dobrovol’skii, N.M., Dobrovolsky, N.N. (2014). On Hyperbolic Zeta Function of Lattices. In: Zgurovsky, M., Sadovnichiy, V. (eds) Continuous and Distributed Systems. Solid Mechanics and Its Applications, vol 211. Springer, Cham. https://doi.org/10.1007/978-3-319-03146-0_2

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  • DOI: https://doi.org/10.1007/978-3-319-03146-0_2

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