Skip to main content

2010 | OriginalPaper | Buchkapitel

Crystal Symmetry Viewed as Zeta Symmetry II

verfasst von : Shigeru Kanemitsu, Haruo Tsukada

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

Verlag: Springer New York

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

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.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

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!

Literatur
1.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Metadaten
Titel
Crystal Symmetry Viewed as Zeta Symmetry II
verfasst von
Shigeru Kanemitsu
Haruo Tsukada
Copyright-Jahr
2010
Verlag
Springer New York
DOI
https://doi.org/10.1007/978-1-4419-6263-8_16