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

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

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UDC: 511.9
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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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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Proc. Steklov Inst. Math., 275 (2011), 229–238
2. P. M. Gruber, “Application of an idea of Vorono\u i to lattice packing”, Ann. Mat. Pura Appl., 193:4 (2014), 939–959
3. R. Coulangeon, G. Lazzarini, “Spherical designs and heights of Euclidean lattices”, J. Number Theory, 141 (2014), 288–315
4. P. M. Gruber, “Normal bundles of convex bodies”, Adv. Math., 254 (2014), 419–453
5. Betermin L., “Two-Dimensional Theta Functions and Crystallization among Bravais Lattices”, SIAM J. Math. Anal., 48:5 (2016), 3236–3269
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