Inventiones mathematicae

, Volume 165, Issue 1, pp 115–151

Minima of Epstein’s Zeta function and heights of flat tori

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References

  1. 1.
    Cassels, J.W.S.: On a problem of Rankin about the Epstein zeta function. Proc. Glasg. Math. Assoc. 4, 73–80 (1959); 6, 116 (1963)MATHMathSciNetGoogle Scholar
  2. 2.
    Chiu, P.: Height of flat tori. Proc. Am. Math. Soc. 125, 723–730 (1997)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Kok Seng Chua: The height of the Leech lattice. Bull. Aust. Math. Soc. 62, 243–251 (2000)Google Scholar
  4. 4.
    Cohn, H., Kumar, A.: Optimality and uniqueness of the Leech lattice among lattices. Preprint math.MG/0403263Google Scholar
  5. 5.
    Cohn, H., Kumar, A.: Universally optimal distribution of points on spheres. Preprint 2004Google Scholar
  6. 6.
    Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press 1985Google Scholar
  7. 7.
    Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. New York: Springer 1999Google Scholar
  8. 8.
    Diananda, P.H.: Notes on two lemmas concerning the Epstein zeta-function. Proc. Glasg. Math. Assoc. 6, 202–204 (1964)MATHMathSciNetGoogle Scholar
  9. 9.
    Duistermaat, J.J., Kolk, J.A.C., Varadarajan, V.S.: Spectra of compact locally symmetric manifolds of negative curvature. Invent. Math. 52, 27–93 (1979)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Ennola, V.: A lemma about the Epstein zeta-function. Proc. Glasg. Math. Assoc. 6, 198–201 (1964)MATHMathSciNetGoogle Scholar
  11. 11.
    Ennola, V.: On a problem about the Epstein zeta function. Proc. Camb. Philos. Soc. 60, 855–875 (1964)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Epstein, P.: Zur Theorie allgemeiner Zetafunctionen, I, II. Math. Ann. 56, 615–644 (1903); 63, 205–216 (1907)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. New York: Academic Press 1978Google Scholar
  14. 14.
    Huffman, W.C., Sloane, N.J.A.: Most primitive groups have messy invariants. Adv. Math. 32, 118–127 (1979)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    James, A.T.: Zonal polynomials of the real positive definite symmetric matrices. Ann. Math. 74, 456–469 (1961)MATHCrossRefGoogle Scholar
  16. 16.
    Littlewood, D.E.: The Theory of Group Characters and Matrix Representations of Groups, 2nd edn. Oxford: Claredon Press 1950Google Scholar
  17. 17.
    Minakshisundaram, S., Pleijel, Å.: Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Can. J. Math. 1, 242–256 (1949)MATHMathSciNetGoogle Scholar
  18. 18.
    Montgomery, H.L.: Minimal theta functions. Glasg. Math. J. 30, 75–85 (1988)MATHCrossRefGoogle Scholar
  19. 19.
    Osgood, B., Phillips, R., Sarnak, P.: Extremals of determinants of Laplacians. J. Funct. Anal. 80, 148–211 (1988)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Rankin, R.A.: A minimum problem for the Epstein zeta function. Proc. Glasg. Math. Assoc. 1, 149–158 (1953)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Rogers, C.A.: Mean values over the space of lattices. Acta Math. 94, 249–287 (1955)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Ryshkov, S.S.: On the question of final ζ-optimality of lattices providing the closest lattice packing of n-dimensional spheres. Sib. Mat. Zh. 14, 1065–1075 (1973). English transl., Sib. Math. J. 14, 743–750 (1974)MATHGoogle Scholar
  23. 23.
    Sarnak, P.: Determinants of Laplacians; Heights and Finiteness. In: Analysis, et cetera, pp. 601–622, ed. by P.H. Rabinowitz, E. Zehnder. Boston, MA: Academic Press 1990Google Scholar
  24. 24.
    Sarnak, P., Strömbergsson, A.: Invent. Math., http://dx.doi.org/10.1007/s00222-005-0488-2 (this web-page contains the computer files associated with the present paper)Google Scholar
  25. 25.
    Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. B 20, 47–87 (1956)MATHMathSciNetGoogle Scholar
  26. 26.
    Siegel, C.L.: A mean value theorem in geometry of numbers. Ann. Math. 46, 340–347 (1945)MATHCrossRefGoogle Scholar
  27. 27.
    Stembridge, J.R.: Computational aspects of root systems, Coxeter groups, and Weyl characters. MSJ Memoirs, vol. 11, pp. 1–38. Tokyo: Math. Soc. 2001 (see also www.math.lsa.umich.edu/∼jrs/maple.html)Google Scholar
  28. 28.
    Temme, N.M.: Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta functions. Math. Comp. 29, 1109–1114 (1975)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Temme, N.M.: The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10, 757–766 (1979)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Terras, A.: The minima of quadratic forms and the behavior of Epstein and Dedekind zeta functions. J. Number Theory 12, 258–272 (1980)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications, II. Berlin: Springer 1988Google Scholar
  32. 32.
    Torquato, S., Stillinger, F.H.: Local density fluctuations, hyperuniform systems, and order metrics. Phys. Rev. E 68, 1–25 (2003)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Advanced Study and Princeton UniversityPrincetonUSA
  2. 2.Department of MathematicsUppsala UniversityUppsalaSweden

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