Skip to main content
Log in

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

  • Published:
Inventiones mathematicae Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  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)

    Article  Google Scholar 

  2. Chiu, P.: Height of flat tori. Proc. Am. Math. Soc. 125, 723–730 (1997)

    Article  MathSciNet  Google Scholar 

  3. Kok Seng Chua: The height of the Leech lattice. Bull. Aust. Math. Soc. 62, 243–251 (2000)

    Article  Google Scholar 

  4. Cohn, H., Kumar, A.: Optimality and uniqueness of the Leech lattice among lattices. Preprint math.MG/0403263

  5. Cohn, H., Kumar, A.: Universally optimal distribution of points on spheres. Preprint 2004

  6. Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press 1985

  7. Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. New York: Springer 1999

  8. Diananda, P.H.: Notes on two lemmas concerning the Epstein zeta-function. Proc. Glasg. Math. Assoc. 6, 202–204 (1964)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  10. Ennola, V.: A lemma about the Epstein zeta-function. Proc. Glasg. Math. Assoc. 6, 198–201 (1964)

    Article  Google Scholar 

  11. Ennola, V.: On a problem about the Epstein zeta function. Proc. Camb. Philos. Soc. 60, 855–875 (1964)

    Article  Google Scholar 

  12. Epstein, P.: Zur Theorie allgemeiner Zetafunctionen, I, II. Math. Ann. 56, 615–644 (1903); 63, 205–216 (1907)

    Article  MathSciNet  Google Scholar 

  13. Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. New York: Academic Press 1978

  14. Huffman, W.C., Sloane, N.J.A.: Most primitive groups have messy invariants. Adv. Math. 32, 118–127 (1979)

    Article  Google Scholar 

  15. James, A.T.: Zonal polynomials of the real positive definite symmetric matrices. Ann. Math. 74, 456–469 (1961)

    Article  MathSciNet  Google Scholar 

  16. Littlewood, D.E.: The Theory of Group Characters and Matrix Representations of Groups, 2nd edn. Oxford: Claredon Press 1950

  17. Minakshisundaram, S., Pleijel, Å.: Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Can. J. Math. 1, 242–256 (1949)

    Article  Google Scholar 

  18. Montgomery, H.L.: Minimal theta functions. Glasg. Math. J. 30, 75–85 (1988)

    Article  Google Scholar 

  19. Osgood, B., Phillips, R., Sarnak, P.: Extremals of determinants of Laplacians. J. Funct. Anal. 80, 148–211 (1988)

    Article  MathSciNet  Google Scholar 

  20. Rankin, R.A.: A minimum problem for the Epstein zeta function. Proc. Glasg. Math. Assoc. 1, 149–158 (1953)

    Article  Google Scholar 

  21. Rogers, C.A.: Mean values over the space of lattices. Acta Math. 94, 249–287 (1955)

    Article  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  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 1990

  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)

  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)

    Google Scholar 

  26. Siegel, C.L.: A mean value theorem in geometry of numbers. Ann. Math. 46, 340–347 (1945)

    Article  MathSciNet  Google Scholar 

  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)

  28. Temme, N.M.: Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta functions. Math. Comp. 29, 1109–1114 (1975)

    Article  MathSciNet  Google Scholar 

  29. Temme, N.M.: The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10, 757–766 (1979)

    Article  MathSciNet  Google Scholar 

  30. Terras, A.: The minima of quadratic forms and the behavior of Epstein and Dedekind zeta functions. J. Number Theory 12, 258–272 (1980)

    Article  MathSciNet  Google Scholar 

  31. Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications, II. Berlin: Springer 1988

  32. Torquato, S., Stillinger, F.H.: Local density fluctuations, hyperuniform systems, and order metrics. Phys. Rev. E 68, 1–25 (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Peter Sarnak or Andreas Strömbergsson.

Additional information

To Audrey Terras on the occasion of her 60th birthday

Electronic supplementary material

The supplementary material for this paper is contained in a zipped directory called minima.zip

To save the files to your computer, simply click the link and select “Save as ...”

Supplementary material, approximately 412 KB.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sarnak, P., Strömbergsson, A. Minima of Epstein’s Zeta function and heights of flat tori. Invent. math. 165, 115–151 (2006). https://doi.org/10.1007/s00222-005-0488-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-005-0488-2

Keywords

Navigation