The permutation group method for the dilogarithm

Georges Rhin[1]; Carlo Viola[2]

  • [1] Département de Mathématiques UFR MIM Université de Metz Ile du Saulcy 57045 Metz Cedex 01, France
  • [2] Dipartimento di Matematica Università di Pisa Largo B. Pontecorvo 5 56127 Pisa, Italy

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2005)

  • Volume: 4, Issue: 3, page 389-437
  • ISSN: 0391-173X

Abstract

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We give qualitative and quantitative improvements on all the best previously known irrationality results for dilogarithms of positive rational numbers. We obtain such improvements by applying our permutation group method to the diophantine study of double integrals of rational functions related to the dilogarithm.

How to cite

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Rhin, Georges, and Viola, Carlo. "The permutation group method for the dilogarithm." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 4.3 (2005): 389-437. <http://eudml.org/doc/84565>.

@article{Rhin2005,
abstract = {We give qualitative and quantitative improvements on all the best previously known irrationality results for dilogarithms of positive rational numbers. We obtain such improvements by applying our permutation group method to the diophantine study of double integrals of rational functions related to the dilogarithm.},
affiliation = {Département de Mathématiques UFR MIM Université de Metz Ile du Saulcy 57045 Metz Cedex 01, France; Dipartimento di Matematica Università di Pisa Largo B. Pontecorvo 5 56127 Pisa, Italy},
author = {Rhin, Georges, Viola, Carlo},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {389-437},
publisher = {Scuola Normale Superiore, Pisa},
title = {The permutation group method for the dilogarithm},
url = {http://eudml.org/doc/84565},
volume = {4},
year = {2005},
}

TY - JOUR
AU - Rhin, Georges
AU - Viola, Carlo
TI - The permutation group method for the dilogarithm
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2005
PB - Scuola Normale Superiore, Pisa
VL - 4
IS - 3
SP - 389
EP - 437
AB - We give qualitative and quantitative improvements on all the best previously known irrationality results for dilogarithms of positive rational numbers. We obtain such improvements by applying our permutation group method to the diophantine study of double integrals of rational functions related to the dilogarithm.
LA - eng
UR - http://eudml.org/doc/84565
ER -

References

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  1. [1] F. Beukers, A note on the irrationality of ζ ( 2 ) and ζ ( 3 ) , Bull. London Math. Soc. 11 (1979), 268-272. Zbl0421.10023MR554391
  2. [2] E. Bombieri, On G -functions, In: “Recent Progress in Analytic Number Theory”, H. Halberstam and C. Hooley (eds.), Durham, 1979, Academic Press, London-New York, 1981, vol. 2, 1-67. Zbl0461.10031MR637359
  3. [3] M. Hata, Rational approximations to the dilogarithm, Trans. Amer. Math. Soc. 336 (1993), 363-387. Zbl0768.11022MR1147401
  4. [4] G. Rhin and C. Viola, On a permutation group related to ζ ( 2 ) , Acta Arith. 77 (1996), 23-56. Zbl0864.11037MR1404975
  5. [5] G. Rhin and C. Viola, The group structure for ζ ( 3 ) , Acta Arith. 97 (2001), 269-293. Zbl1004.11042MR1826005
  6. [6] C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss., 1929, n. 1; Gesammelte Abhandlungen, Band I, Springer-Verlag, Berlin-Heidelberg, 1966, 209-266. Zbl56.0180.05JFM56.0180.05
  7. [7] C. Viola, On Siegel’s method in diophantine approximation to transcendental numbers, Rend. Sem. Mat. Univ. Pol. Torino 53 (1995), 455-469. Zbl0873.11043MR1452398

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