Arithmetic of linear forms involving odd zeta values

Wadim Zudilin[1]

  • [1] Department of Mechanics and Mathematics Moscow Lomonosov State University Vorobiovy Gory, GSP-2 119992 Moscow, Russia

Journal de Théorie des Nombres de Bordeaux (2004)

  • Volume: 16, Issue: 1, page 251-291
  • ISSN: 1246-7405

Abstract

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A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of ζ ( 2 ) and ζ ( 3 ) , as well as to explain Rivoal’s recent result on infiniteness of irrational numbers in the set of odd zeta values, and to prove that at least one of the four numbers ζ ( 5 ) , ζ ( 7 ) , ζ ( 9 ) , and ζ ( 11 ) is irrational.

How to cite

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Zudilin, Wadim. "Arithmetic of linear forms involving odd zeta values." Journal de Théorie des Nombres de Bordeaux 16.1 (2004): 251-291. <http://eudml.org/doc/249254>.

@article{Zudilin2004,
abstract = {A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of $\zeta (2)$ and $\zeta (3)$, as well as to explain Rivoal’s recent result on infiniteness of irrational numbers in the set of odd zeta values, and to prove that at least one of the four numbers $\zeta (5)$, $\zeta (7)$, $\zeta (9)$, and $\zeta (11)$ is irrational.},
affiliation = {Department of Mechanics and Mathematics Moscow Lomonosov State University Vorobiovy Gory, GSP-2 119992 Moscow, Russia},
author = {Zudilin, Wadim},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {irrationality; zeta values; group structure},
language = {eng},
number = {1},
pages = {251-291},
publisher = {Université Bordeaux 1},
title = {Arithmetic of linear forms involving odd zeta values},
url = {http://eudml.org/doc/249254},
volume = {16},
year = {2004},
}

TY - JOUR
AU - Zudilin, Wadim
TI - Arithmetic of linear forms involving odd zeta values
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2004
PB - Université Bordeaux 1
VL - 16
IS - 1
SP - 251
EP - 291
AB - A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of $\zeta (2)$ and $\zeta (3)$, as well as to explain Rivoal’s recent result on infiniteness of irrational numbers in the set of odd zeta values, and to prove that at least one of the four numbers $\zeta (5)$, $\zeta (7)$, $\zeta (9)$, and $\zeta (11)$ is irrational.
LA - eng
KW - irrationality; zeta values; group structure
UR - http://eudml.org/doc/249254
ER -

References

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  1. R. Apéry, Irrationalité de ζ ( 2 ) et ζ ( 3 ) . Astérisque 61 (1979), 11–13. Zbl0401.10049
  2. W. N. Bailey, Some transformations of generalized hypergeometric series, and contour integrals of Barnes’s type. Quart. J. Math. Oxford 3:11 (1932), 168–182. Zbl0005.40001
  3. W. N. Bailey, Transformations of well-poised hypergeometric series. Proc. London Math. Soc. II Ser. 36:4 (1934), 235–240. Zbl0008.07101
  4. W. N. Bailey, Generalized hypergeometric series. Cambridge Math. Tracts 32 (Cambridge University Press, Cambridge, 1935); 2nd reprinted edition (Stechert-Hafner, New York, NY, 1964). Zbl0011.02303MR185155
  5. K. Ball, T. Rivoal, Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs. Invent. Math. 146:1 (2001), 193–207. Zbl1058.11051MR1859021
  6. F. Beukers, A note on the irrationality of  ζ ( 2 ) and  ζ ( 3 ) . Bull. London Math. Soc. 11:3 (1979), 268–272. Zbl0421.10023MR554391
  7. G. V. Chudnovsky, On the method of Thue–Siegel. Ann. of Math. II Ser. 117:2 (1983), 325–382. Zbl0518.10038MR690849
  8. R. Dvornicich, C. Viola, Some remarks on Beukers’ integrals. Colloq. Math. Soc. János Bolyai 51 (North-Holland, Amsterdam, 1987), 637–657. Zbl0755.11019MR1058238
  9. N. I. Fel’dman, Yu. V. Nesterenko, Transcendental numbers. (Number theory IV), Encyclopaedia Math. Sci. 44 (Springer-Verlag, Berlin, 1998). Zbl0885.11004MR1603608
  10. L. A. Gutnik, On the irrationality of certain quantities involving  ζ ( 3 ) . Uspekhi Mat. Nauk [Russian Math. Surveys] 34:3 (1979), 190; Acta Arith. 42:3 (1983), 255–264. Zbl0437.10015MR729735
  11. M. Hata, Legendre type polynomials and irrationality measures. J. Reine Angew. Math. 407:1 (1990), 99–125. Zbl0692.10034MR1048530
  12. M. Hata, Irrationality measures of the values of hypergeometric functions. Acta Arith. 60:4 (1992), 335–347. Zbl0760.11020MR1159350
  13. M. Hata, Rational approximations to the dilogarithm. Trans. Amer. Math. Soc. 336:1 (1993), 363–387. Zbl0768.11022MR1147401
  14. M. Hata, A note on Beukers’ integral. J. Austral. Math. Soc. Ser. A 58:2 (1995), 143–153. Zbl0830.11026MR1323987
  15. M. Hata, A new irrationality measure for  ζ ( 3 ) . Acta Arith. 92:1 (2000), 47–57. Zbl0955.11023MR1739738
  16. A. Heimonen, T. Matala-Aho, K. Väänänen, On irrationality measures of the values of Gauss hypergeometric function. Manuscripta Math. 81:1/2 (1993), 183–202. Zbl0801.11032MR1247597
  17. T. G. Hessami Pilerhood, Arithmetic properties of values of hypergeometric functions. Ph. D. thesis (Moscow University, Moscow, 1999); Linear independence of vectors with polylogarithmic coordinates. Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] 6 (1999), 54–56. Zbl0983.11044
  18. Yu. L. Luke, Mathematical functions and their approximations. (Academic Press, New York, NY, 1975). Zbl0318.33001MR501762
  19. Yu. V. Nesterenko, A few remarks on  ζ ( 3 ) . Mat. Zametki [Math. Notes] 59:6 (1996), 865–880. Zbl0888.11028MR1445472
  20. Yu. V. Nesterenko, Integral identities and constructions of approximations to zeta values. Actes des 12èmes rencontres arithmétiques de Caen (June 29–30, 2001), J. Théorie Nombres Bordeaux 15:2 (2003), 535–550. Zbl1090.11047MR2140866
  21. Yu. V. Nesterenko, Arithmetic properties of values of the Riemann zeta function and generalized hypergeometric functions. Manuscript (2001). 
  22. E. M. Nikishin, On irrationality of values of functions F ( x , s ) . Mat. Sb. [Russian Acad. Sci. Sb. Math.] 109:3 (1979), 410–417. Zbl0414.10032MR542809
  23. A. van der Poorten, A proof that Euler missed... Apéry’s proof of the irrationality of  ζ ( 3 ) . An informal report, Math. Intelligencer 1:4 (1978/79), 195–203. Zbl0409.10028MR547748
  24. G. Rhin, C. Viola, On the irrationality measure of  ζ ( 2 ) . Ann. Inst. Fourier (Grenoble) 43:1 (1993), 85–109. Zbl0776.11036MR1209696
  25. G. Rhin, C. Viola, On a permutation group related to  ζ ( 2 ) . Acta Arith. 77:1 (1996), 23–56. Zbl0864.11037MR1404975
  26. G. Rhin, C. Viola, The group structure for  ζ ( 3 ) . Acta Arith. 97:3 (2001), 269–293. Zbl1004.11042MR1826005
  27. T. Rivoal, La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs. C. R. Acad. Sci. Paris Sér. I Math. 331:4 (2000), 267–270. Zbl0973.11072MR1787183
  28. T. Rivoal, Irrationnalité d’une infinité de valeurs de la fonction zêta aux entiers impairs. Rapport de recherche SDAD no. 2000-9 (Université de Caen, Caen, 2000). 
  29. T. Rivoal, Propriétés diophantiennes des valeurs de la fonction zêta de Riemann aux entiers impairs. Thèse de doctorat (Université de Caen, Caen, 2001). 
  30. T. Rivoal, Irrationalité d’au moins un des neuf nombres ζ ( 5 ) , ζ ( 7 ) , ⋯ , ζ ( 21 ) . Acta Arith. 103 (2001), 157–167. Zbl1015.11033MR1904870
  31. E. A. Rukhadze, A lower bound for the approximation of  ln 2 by rational numbers. Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] 6 (1987), 25–29. Zbl0635.10025MR922879
  32. L. J. Slater, Generalized hypergeometric functions. 2nd edition (Cambridge University Press, Cambridge, 1966). Zbl0135.28101MR201688
  33. D. V. Vasilyev, On small linear forms for the values of the Riemann zeta-function at odd points. Preprint no. 1 (558) (Nat. Acad. Sci. Belarus, Institute Math., Minsk, 2001). 
  34. C. Viola, Hypergeometric functions and irrationality measures. Analytic Number Theory (ed. Y. Motohashi), London Math. Soc. Lecture Note Ser. 247 (Cambridge University Press, Cambridge, 1997), 353–360. Zbl0904.11020MR1695002
  35. W. V. Zudilin, Irrationality of values of zeta function at odd integers. Uspekhi Mat. Nauk [Russian Math. Surveys] 56:2 (2001), 215–216. Zbl1037.11048MR1859714
  36. W. Zudilin, Irrationality of values of zeta-function. Contemporary research in mathematics and mechanics, Proceedings of the 23rd Conference of Young Scientists of the Department of Mechanics and Mathematics (Moscow State University, April 9–14, 2001), part 2 (Publ. Dept. Mech. Math. MSU, Moscow, 2001), 127–135; E-print math.NT/0104249. 
  37. W. Zudilin, Irrationality of values of Riemann’s zeta function. Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.] 66:3 (2002), 49–102. Zbl1114.11305MR1921809
  38. W. V. Zudilin, One of the eight numbers ζ ( 5 ) , ζ ( 7 ) , ⋯ , ζ ( 17 ) , ζ ( 19 ) is irrational. Mat. Zametki [Math. Notes] 70:3 (2001), 472–476. Zbl1022.11035MR1882257
  39. W. V. Zudilin, Cancellation of factorials. Mat. Sb. [Russian Acad. Sci. Sb. Math.] 192:8 (2001), 95–122. Zbl1030.11032MR1862246
  40. W. Zudilin, Well-poised hypergeometric service for diophantine problems of zeta values. Actes des 12èmes rencontres arithmétiques de Caen (June 29–30, 2001), J. Théorie Nombres Bordeaux 15:2 (2003), 593–626. Zbl1156.11326MR2140869

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