Skip to main content
Top

2010 | OriginalPaper | Chapter

Crystal Symmetry Viewed as Zeta Symmetry II

Authors : Shigeru Kanemitsu, Haruo Tsukada

Published in: The Legacy of Alladi Ramakrishnan in the Mathematical Sciences

Publisher: Springer New York

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Summary

In this paper, we continue our previous investigations on applications of the Epstein zeta-functions. We shall mostly state the results for the lattice zeta-functions, which can be immediately translated into those for the corresponding Epstein zeta-functions. We shall take up the generalized Chowla–Selberg (integral) formula and state many concrete special cases of this formula.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference B. C. Berndt, Identities involving the coefficients of a class of Dirichlet series IV, Trans. Am. Math. Soc. 149 (1970), 179–185. B. C. Berndt, Identities involving the coefficients of a class of Dirichlet series IV, Trans. Am. Math. Soc. 149 (1970), 179–185.
2.
go back to reference B. C. Berndt, Identities involving the coefficients of a class of Dirichlet series VI, ibid, 160 (1971), 157–167. B. C. Berndt, Identities involving the coefficients of a class of Dirichlet series VI, ibid, 160 (1971), 157–167.
3.
go back to reference J. M. Borwein and P. B. Borwein, Pi and the AGM: A study in analytic number theory and computational complexity, Wiley, New York, (1987).MATH J. M. Borwein and P. B. Borwein, Pi and the AGM: A study in analytic number theory and computational complexity, Wiley, New York, (1987).MATH
4.
go back to reference A. N. Chaba and R. K. Pathria, Evaluation of a class of lattice sums in arbitrary dimensions, J. Math. Phys. 16 (1975), 1457–1460. A. N. Chaba and R. K. Pathria, Evaluation of a class of lattice sums in arbitrary dimensions, J. Math. Phys. 16 (1975), 1457–1460.
5.
go back to reference A. N. Chaba and R. K. Pathria, Evaluation of a class of lattice sums using Poisson’s summation formula. II, J. Phys. A: Math. Gen. 9 (1976), 1411–1423. A. N. Chaba and R. K. Pathria, Evaluation of a class of lattice sums using Poisson’s summation formula. II, J. Phys. A: Math. Gen. 9 (1976), 1411–1423.
6.
go back to reference S. Chowla and A. Selberg, On Epstein’s zeta-function (I), Proc. Nat. Acad. Sci. USA 35 (1949), 371–374; Collected Papers of Atle Selberg I, Springer Verlag, (1989), 367–370. The Collected Papers of Sarvadaman Chowla II, CRM, (1999), 719–722. S. Chowla and A. Selberg, On Epstein’s zeta-function (I), Proc. Nat. Acad. Sci. USA 35 (1949), 371–374; Collected Papers of Atle Selberg I, Springer Verlag, (1989), 367–370. The Collected Papers of Sarvadaman Chowla II, CRM, (1999), 719–722.
7.
go back to reference A. Selberg and S. Chowla, On Epstein’s zeta-function, J. Reine Angew, Math. 227 (1967), 86–110; Collected Papers of Atle Selberg I, Springer Verlag, (1989), 521–545; The Collected Papers of Sarvadaman Chowla II, CRM, (1999), 1101–1125. A. Selberg and S. Chowla, On Epstein’s zeta-function, J. Reine Angew, Math. 227 (1967), 86–110; Collected Papers of Atle Selberg I, Springer Verlag, (1989), 521–545; The Collected Papers of Sarvadaman Chowla II, CRM, (1999), 1101–1125.
8.
go back to reference J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (2nd. ed.), Springer, New York, (1993). J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (2nd. ed.), Springer, New York, (1993).
9.
go back to reference R. E. Crandall, New representations for the Madelung constant, Exp. Math. 8 (1999), 367–379. R. E. Crandall, New representations for the Madelung constant, Exp. Math. 8 (1999), 367–379.
10.
go back to reference P. Ewald, Zur Theorie allgemeiner Zetafunctionen II, Ann. Phys. 63 (1921), 205–216. P. Ewald, Zur Theorie allgemeiner Zetafunctionen II, Ann. Phys. 63 (1921), 205–216.
11.
go back to reference M. L. Glasser, The evaluation of lattice sums I: Analytic procedures, J. Math. Phys. 14 (1973), 409–413; Comments by A. Hautot, ibid. 15 (1984), 268. M. L. Glasser, The evaluation of lattice sums I: Analytic procedures, J. Math. Phys. 14 (1973), 409–413; Comments by A. Hautot, ibid. 15 (1984), 268.
12.
go back to reference M. L. Glasser and I. J. Zucker, Lattice sums,, Theoretical Chemistry: Advances and Perspectives, Vol. 5, ed. by D. Henderson, Academic, New York (1980), 67–139. M. L. Glasser and I. J. Zucker, Lattice sums,, Theoretical Chemistry: Advances and Perspectives, Vol. 5, ed. by D. Henderson, Academic, New York (1980), 67–139.
13.
go back to reference G. H. Hardy, Some multiple integrals, Quart. J. Math. (Oxford)(2) 5 (1908), 357–375; Collected Papers. Vol. V (1972), 434–452, Comments 453. G. H. Hardy, Some multiple integrals, Quart. J. Math. (Oxford)(2) 5 (1908), 357–375; Collected Papers. Vol. V (1972), 434–452, Comments 453.
14.
go back to reference A. Hautot, A new method for the evaluation of slowly convergent series, J. Math. Phys. 15 (1974), 1722–1727. A. Hautot, A new method for the evaluation of slowly convergent series, J. Math. Phys. 15 (1974), 1722–1727.
15.
go back to reference A. Hautot, New applications of Poisson’s summation formula, J. Phys. A Math. Gen. 8 (1975), 853–862. A. Hautot, New applications of Poisson’s summation formula, J. Phys. A Math. Gen. 8 (1975), 853–862.
16.
go back to reference S. Kanemitsu and H. Tsukada, Vistas of special functions, World Scientific, Singapore (2007), pp. 215.MATHCrossRef S. Kanemitsu and H. Tsukada, Vistas of special functions, World Scientific, Singapore (2007), pp. 215.MATHCrossRef
17.
go back to reference S. Kanemitsu, Y. Tanigawa, H. Tsukada and M. Yoshimoto, On Bessel series expressions for some lattice sums II, J. Phys. A Math. Gen. 37 (2004), 719–734. S. Kanemitsu, Y. Tanigawa, H. Tsukada and M. Yoshimoto, On Bessel series expressions for some lattice sums II, J. Phys. A Math. Gen. 37 (2004), 719–734.
18.
go back to reference S. Kanemitsu, Y. Tanigawa and H. Tsukada, Crystal symmetry viewed as zeta symmetry, Proc. Intern. Sympos. Zeta-functions, Topology and Quantum Physics, Kluwer Academic, Dordrecht (2005), 91–129. S. Kanemitsu, Y. Tanigawa and H. Tsukada, Crystal symmetry viewed as zeta symmetry, Proc. Intern. Sympos. Zeta-functions, Topology and Quantum Physics, Kluwer Academic, Dordrecht (2005), 91–129.
19.
go back to reference S. Kanemitsu, Y. Tanigawa and W.-P. Zhang, On Bessel series expressions for some lattice sums, Chebyshevskii Sb. 5 (2004), 128–137.MathSciNetMATH S. Kanemitsu, Y. Tanigawa and W.-P. Zhang, On Bessel series expressions for some lattice sums, Chebyshevskii Sb. 5 (2004), 128–137.MathSciNetMATH
20.
go back to reference M. Katsurada, An application of Mellin–Barnes type of integrals to the mean square of L-functions, Liet. Matem. Rink. 38 (1998), 98–112. M. Katsurada, An application of Mellin–Barnes type of integrals to the mean square of L-functions, Liet. Matem. Rink. 38 (1998), 98–112.
21.
go back to reference A. F. Lavrik, An approximate functional equation for the Dirichlet L-function, Trudy Moskov. Math. Obšč 18 (1968), 91–104=Trans. Moskow Math. Soc. 18 (1968), 101–115. A. F. Lavrik, An approximate functional equation for the Dirichlet L-function, Trudy Moskov. Math. Obšč 18 (1968), 91–104=Trans. Moskow Math. Soc. 18 (1968), 101–115.
22.
go back to reference K. Matsumoto, Recent developments in the mean square theory of the Riemann zeta and other zeta-functions, in Number Theory ed. by R. P. Bambah et al., Hindustan Books Agency, (2000) 241–286. K. Matsumoto, Recent developments in the mean square theory of the Riemann zeta and other zeta-functions, in Number Theory ed. by R. P. Bambah et al., Hindustan Books Agency, (2000) 241–286.
23.
go back to reference R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, Cambridge, (2001).MATHCrossRef R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, Cambridge, (2001).MATHCrossRef
24.
go back to reference A. Terras, Bessel series expansions of the Epstein zeta function and the functional equation, Trans. Am. Math. Soc., 183, (1973) 477–486.MathSciNetMATHCrossRef A. Terras, Bessel series expansions of the Epstein zeta function and the functional equation, Trans. Am. Math. Soc., 183, (1973) 477–486.MathSciNetMATHCrossRef
25.
go back to reference A. Terras, Harmonic Analysis on Symmetric Spaces and Applications I, Springer, New York, (1985).MATHCrossRef A. Terras, Harmonic Analysis on Symmetric Spaces and Applications I, Springer, New York, (1985).MATHCrossRef
26.
go back to reference G. N. Watson, A treatise on the theory of Bessel function, second edition, CUP, Cambridge, (1966). G. N. Watson, A treatise on the theory of Bessel function, second edition, CUP, Cambridge, (1966).
27.
go back to reference I. J. Zucker, Exact results for some lattice sums in 2, 4, 6 and 8 dimensions, J. Phys. A Math. Nucl. Gen. 7 (1974), 1568–1575. I. J. Zucker, Exact results for some lattice sums in 2, 4, 6 and 8 dimensions, J. Phys. A Math. Nucl. Gen. 7 (1974), 1568–1575.
28.
go back to reference I. J. Zucker, Functional equation for poly-dimensional zeta functions and the evaluation of Madelung constants, J. Phys. A Math. Gen. 9 (1976), 499–505. I. J. Zucker, Functional equation for poly-dimensional zeta functions and the evaluation of Madelung constants, J. Phys. A Math. Gen. 9 (1976), 499–505.
Metadata
Title
Crystal Symmetry Viewed as Zeta Symmetry II
Authors
Shigeru Kanemitsu
Haruo Tsukada
Copyright Year
2010
Publisher
Springer New York
DOI
https://doi.org/10.1007/978-1-4419-6263-8_16

Premium Partner