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Tr. Mat. Inst. Steklova, 2012, Volume 276, Pages 109–130 (Mi tm3359)  

This article is cited in 5 scientific papers (total in 5 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

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.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2012, 276, 103–124

Bibliographic databases:

UDC: 511.9
Received in July 2011
Language:

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, Tr. 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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\paper Application of an idea of Vorono\u\i\ to lattice zeta functions
\inbook Number theory, algebra, and analysis
\bookinfo Collected papers. Dedicated to Professor Anatolii Alekseevich Karatsuba on the occasion of his 75th birthday
\serial Tr. Mat. Inst. Steklova
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\vol 276
\pages 109--130
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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\pages 103--124
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Proc. Steklov Inst. Math., 275 (2011), 229–238  mathnet  crossref  mathscinet  isi  elib  elib
    2. 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
    3. R. Coulangeon, G. Lazzarini, “Spherical designs and heights of Euclidean lattices”, J. Number Theory, 141 (2014), 288–315  crossref  mathscinet  zmath  isi  scopus
    4. P. M. Gruber, “Normal bundles of convex bodies”, Adv. Math., 254 (2014), 419–453  crossref  mathscinet  zmath  isi  scopus
    5. 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
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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