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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2012, Volume 276, Pages 109–130 (Mi tm3359)  
This article is cited in 11 scientific papers (total in 11 papers)

Application of an idea of Voronoĭ to lattice zeta functions

Peter M. Gruber

Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Vienna, Austria
Full-text PDF (267 kB) Citations (11)
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Abstract: A major problem in the geometry of numbers is the investigation of the local minima of the Epstein zeta function. In this article refined minimum properties of the Epstein zeta function and more general lattice zeta functions are studied. Using an idea of Voronoĭ, characterizations and sufficient conditions are given for lattices at which the Epstein zeta function is stationary or quadratic minimum. Similar problems of a duality character are investigated for the product of the Epstein zeta function of a lattice and the Epstein zeta function of the polar lattice. Besides Voronoĭ type notions such as versions of perfection and eutaxy, these results involve spherical designs and automorphism groups of lattices. Several results are extended to more general lattice zeta functions, where the Euclidean norm is replaced by a smooth norm.
Received in July 2011
English version:
Proceedings of the Steklov Institute of Mathematics, 2012, Volume 276, Pages 103–124
DOI: https://doi.org/10.1134/S0081543812010099
Bibliographic databases:
Document Type: Article
UDC: 511.9
Language: English
Citation: Peter M. Gruber, “Application of an idea of Voronoĭ to lattice zeta functions”, Number theory, algebra, and analysis, Collected papers. Dedicated to Professor Anatolii Alekseevich Karatsuba on the occasion of his 75th birthday, Trudy Mat. Inst. Steklova, 276, MAIK Nauka/Interperiodica, Moscow, 2012, 109–130; Proc. Steklov Inst. Math., 276 (2012), 103–124
Citation in format AMSBIB
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\publ MAIK Nauka/Interperiodica
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  • https://www.mathnet.ru/eng/tm/v276/p109
    • This publication is cited in the following 11 articles:
    1. Laurent Bétermin, Ladislav Šamaj, Igor Travěnec, “Three‐dimensional lattice ground states for Riesz and Lennard‐Jones–type energies”, Stud Appl Math, 150:1 (2023), 69  crossref
    2. Mathieu Lewin, “Coulomb and Riesz gases: The known and the unknown”, Journal of Mathematical Physics, 63:6 (2022)  crossref
    3. Betermin L., “Theta Functions and Optimal Lattices For a Grid Cells Model”, SIAM J. Appl. Math., 81:5 (2021), 1931–1953  crossref  mathscinet  isi
    4. Betermin L., Faulhuber M., Knuepfer H., “On the Optimality of the Rock-Salt Structure Among Lattices With Charge Distributions”, Math. Models Meth. Appl. Sci., 31:02 (2021), 293–325  crossref  mathscinet  isi
    5. Betermin L., “On Energy Ground States Among Crystal Lattice Structures With Prescribed Bonds”, J. Phys. A-Math. Theor., 54:24 (2021), 245202  crossref  mathscinet  isi
    6. Betermin L., “Minimizing Lattice Structures For Morse Potential Energy in Two and Three Dimensions”, J. Math. Phys., 60:10 (2019), 102901  crossref  mathscinet  isi
    7. Betermin L., “Two-Dimensional Theta Functions and Crystallization among Bravais Lattices”, SIAM J. Math. Anal., 48:5 (2016), 3236–3269  crossref  mathscinet  zmath  isi  scopus
    8. P. M. Gruber, “Application of an idea of Vorono\u i to lattice packing”, Ann. Mat. Pura Appl., 193:4 (2014), 939–959  crossref  mathscinet  zmath  isi  scopus
    9. R. Coulangeon, G. Lazzarini, “Spherical designs and heights of Euclidean lattices”, J. Number Theory, 141 (2014), 288–315  crossref  mathscinet  zmath  isi  scopus
    10. P. M. Gruber, “Normal bundles of convex bodies”, Adv. Math., 254 (2014), 419–453  crossref  mathscinet  zmath  isi  scopus
    11. Proc. Steklov Inst. Math., 275 (2011), 229–238  mathnet  crossref  mathscinet  isi  elib  elib
    Citing articles in Google Scholar: Russian citations, English citations
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    		Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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