Well-poised hypergeometric service for diophantine problems of zeta values

Wadim Zudilin

Journal de théorie des nombres de Bordeaux (2003)

  • Volume: 15, Issue: 2, page 593-626
  • ISSN: 1246-7405

Abstract

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It is explained how the classical concept of well-poised hypergeometric series and integrals becomes crucial in studying arithmetic properties of the values of Riemann’s zeta function. By these well-poised means we obtain: (1) a permutation group for linear forms in 1 and ζ ( 4 ) = π 4 / 90 yielding a conditional upper bound for the irrationality measure of ζ ( 4 ) ; (2) a second-order Apéry-like recursion for ζ ( 4 ) and some low-order recursions for linear forms in odd zeta values; (3) a rich permutation group for a family of certain Euler-type multiple integrals that generalize so-called Beukers’ integrals for ζ ( 2 ) and ζ ( 3 ) .

How to cite

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Zudilin, Wadim. "Well-poised hypergeometric service for diophantine problems of zeta values." Journal de théorie des nombres de Bordeaux 15.2 (2003): 593-626. <http://eudml.org/doc/249095>.

@article{Zudilin2003,
abstract = {It is explained how the classical concept of well-poised hypergeometric series and integrals becomes crucial in studying arithmetic properties of the values of Riemann’s zeta function. By these well-poised means we obtain: (1) a permutation group for linear forms in $1$ and $\zeta (4) = \pi ^4/90$ yielding a conditional upper bound for the irrationality measure of $\zeta (4)$; (2) a second-order Apéry-like recursion for $\zeta (4)$ and some low-order recursions for linear forms in odd zeta values; (3) a rich permutation group for a family of certain Euler-type multiple integrals that generalize so-called Beukers’ integrals for $\zeta (2)$ and $\zeta (3)$.},
author = {Zudilin, Wadim},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {well-poised hypergeometric series},
language = {eng},
number = {2},
pages = {593-626},
publisher = {Université Bordeaux I},
title = {Well-poised hypergeometric service for diophantine problems of zeta values},
url = {http://eudml.org/doc/249095},
volume = {15},
year = {2003},
}

TY - JOUR
AU - Zudilin, Wadim
TI - Well-poised hypergeometric service for diophantine problems of zeta values
JO - Journal de théorie des nombres de Bordeaux
PY - 2003
PB - Université Bordeaux I
VL - 15
IS - 2
SP - 593
EP - 626
AB - It is explained how the classical concept of well-poised hypergeometric series and integrals becomes crucial in studying arithmetic properties of the values of Riemann’s zeta function. By these well-poised means we obtain: (1) a permutation group for linear forms in $1$ and $\zeta (4) = \pi ^4/90$ yielding a conditional upper bound for the irrationality measure of $\zeta (4)$; (2) a second-order Apéry-like recursion for $\zeta (4)$ and some low-order recursions for linear forms in odd zeta values; (3) a rich permutation group for a family of certain Euler-type multiple integrals that generalize so-called Beukers’ integrals for $\zeta (2)$ and $\zeta (3)$.
LA - eng
KW - well-poised hypergeometric series
UR - http://eudml.org/doc/249095
ER -

References

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  1. [A1] Yu M. Aleksentsev, On the measure of approximation for the number π by algebraic numbers. Mat. Zametki [Math. Notes]66:4 (1999), 483-493. 
  2. [An] G.E. Andrews, The well-poised thread: An organized chronicle of some amazing summations and their implications. The Ramanujan J.1:1 (1997), 7-23. Zbl0934.11050MR1607525
  3. [Ap] R. Apéry, Irrationalité de ζ(2) et ζ(3). Astérisque61 (1979), 11-13. Zbl0401.10049
  4. [Ba] W.N. Bailey, Generalized hypergeometric series. Cambridge Math. Tracts32 (Cambridge Univ. Press, Cambridge, 1935); 2nd reprinted edition Stechert-Hafner, New York-London, 1964. Zbl0011.02303MR185155JFM61.0406.01
  5. [BR] 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. [Be1] F. Beukers, A note on the irrationality of ζ(2) and ζ(3). Bull. London Math. Soc. 11:3 (1979), 268-272. Zbl0421.10023
  7. [Be2] F. Beukers, Padé approximations in number theory. Lecture Notes in Math.888, Springer-Verlag, Berlin, 1981, 90-99. Zbl0478.10016MR649087
  8. [Be3] F. Beukers, Irrationality proofs using modular forms. Astérisque147-148 (1987), 271-283. Zbl0613.10031MR891433
  9. [Be4] F. Beukers, On Dwork's accessory parameter problem. Math. Z.241:2 (2002), 425-444. Zbl1023.34081MR1935494
  10. [Br] N.G. De Bruijn, Asymptotic methods in analysis. North-Holland Publ., Amsterdam, 1958. Zbl0082.04202
  11. [Co] H. Cohen, Accélération de la convergence de certaines récurrences linéaires, Séminaire de Théorie des nombres de Bordeaux (Année 1980-81), exposé 16, 2 pages. Zbl0479.10023
  12. [Gu] 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. Zbl0437.10015
  13. [Han] J. Hancl, A simple proof of the irrationality of π4. Amer. Math. Monthly93 (1986), 374-375. Zbl0597.10029
  14. [Hat] M. Hata, Legendre type polynomials and irrationality measures. J. Reine Angew. Math.407:1 (1990), 99-125. Zbl0692.10034MR1048530
  15. [JT] W.B. Jones, W.J. Thron, Continued fractions. Analytic theory and applications. Encyclopaedia Math. Appl. Section: Analysis11, Addison-Wesley, London, 1980. Zbl0603.30009MR595864
  16. [Nel] Yu V. Nesterenko, A few remarks on ζ(3). Mat. Zametki [Math. Notes]59:6 (1996), 865-880. Zbl0888.11028
  17. [Ne2] Yu V. Nesterenko, Integral identities and constructions of approximations to zeta val-. ues. J. Théor. Nombres Bordeaux15 (2003), ?-?. Zbl1090.11047MR2140866
  18. [Ne3] Yu V. Nesterenko, Arithmetic properties of values of the Riemann zeta function and generalized hypergeometric functions in preparation (2002). 
  19. [PWZ] M. Petkovšek, H.S. Wilf, D. Zeilberger, A = B. A. K. Peters, Ltd., Wellesley, MA, 1997. MR1379802
  20. [Po] A. Van Der Poorten, A proof that Euler missed... Apéry's proof of the irrationality of ζ(3). An informal report, Math. Intelligencer1:4 (1978/79), 195-203. Zbl0409.10028
  21. [RV1] G. Rhin, C. Viola, On a permutation group related to ζ(2). Acta Arith.77:1 (1996), 23-56. Zbl0864.11037
  22. [RV2] G. Rhin, C. Viola, The group structure for ζ(3). Acta Arith.97:3 (2001), 269-293. Zbl1004.11042
  23. [Ri1] 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
  24. [Ri2] T. Rivoal, Propriétés diophantiennes des valeurs de la fonction zêta de Riemann aux entiers impairs. Thèse de Doctorat, Univ. de Caen, 2001. 
  25. [Ri3] T. Rivoal, Séries hypergéométriques et irrationalité des valeurs de la fonction zêta. J. Théor. Nombres Bordeaux15 (2003), 351-365. Zbl1041.11051MR2019020
  26. [So1] V.N. Sorokin, Hermite-Padé approximations for Nikishin's systems and irrationality of ζ(3). Uspekhi Mat. Nauk [Russian Math. Surveys]49:2 (1994), 167-168. Zbl0827.11042
  27. [So2] V.N. Sorokin, A transcendence measure of π2. Mat. Sb. [Russian Acad. Sci. Sb. Math.] 187:12 (1996), 87-120. Zbl0876.11035
  28. [So3] V.N. Sorokin, Apéry's theorem. Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] 53:3 (1998), 48-52. Zbl1061.11501MR1708549
  29. [So4] V.N. Sorokin, One algorithm for fast calculation of π4. Preprint (Russian Academy of Sciences, M. V. Keldysh Institute for Applied Mathematics, Moscow, 2002), 59 pages; http://www.wis.kuleuven.ac.be/applied/intas/Art5.pdf. 
  30. [VaO] O.N. Vasilenko, Certain formulae for values of the Riemann zeta-function at integral points. Number theory and its applications, Proceedings of the science-theoretic conference (Tashkent, September 26-28, 1990), 27 (Russian). 
  31. [VaD] 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). 
  32. [Vi] C. Viola, Birational transformations and values of the Riemann zeta-function. J. Théor. Nombres Bordeaux15 (2003), ?-? Zbl1074.11041MR2140868
  33. [WZ] H.S. Wilf, D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities. Invent. Math.108:3 (1992), 575-633. Zbl0739.05007MR1163239
  34. [Zl1] S.A. Zlobin, Integrals expressible as linear forms in generalized polylogarithms. Mat. Zametki [Math. Notes]71:5 (2002), 782-787. Zbl1049.11077MR1936201
  35. [Zl2] S.A. Zlobin, On some integral identities. Uspekhi Mat. Nauk [Russian Math. Surveys]57:3 (2002), 153-154. Zbl1058.11057MR1918865
  36. [Zu1] W. Zudilin, Difference equations and the irrationality measure of numbers. Collection of papers: Analytic number theory and applications, Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.]218 (1997), 165-178. Zbl0910.11032MR1642377
  37. [Zu2] 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. [Zu3] W.V. Zudilin, One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational. Uspekhi Mat. Nauk [Russian Math. Surveys]56:4 (2001), 149-150. Zbl1047.11072
  39. [Zu4] W. Zudilin, Arithmetic of linear forms involving odd zeta values. J. Théor. Nombres Bordeaux, to appear. Zbl1156.11327MR2145585
  40. [Zu5] W. Zudilin, An elementary proof of Apérys theorem. E-print math.NT/0202159 (February 2002). 

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