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

top
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

top

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

top
  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). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.