Zeta values on the Web

Several references and links devoted to the arithmetic study of values of the Riemann zeta function at positive integers and related constants are put on this page.

The following list has not been updated for many years but may get some new boost in the future.


  1. F. Lindemann, Über die Zalh π, Math. Annalen 20 (1882), 213--225 (EMIS link)


  2. R. Apéry, Irrationalité de ζ(2) et ζ(3), Astérisque 61 (1979), 11--13
  3. F. Beukers, A note on the irrationality of ζ(2) and ζ(3), Bull. London Math. Soc. 11:3 (1979), 268--272 (Russian version as pdf, gzip ps)
  4. H. Cohen, Démonstration de l'irrationalité de ζ(3) (d'après Apéry), Séminaire de Théorie des Nombres de Grenoble VI (1978), 1--9
  5. L.A. Gutnik, On the irrationality of certain quantities involving ζ(3), Uspekhi Mat. Nauk [Russian Math. Surveys] 34:3 (1979), 190
  6. 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 (html)
  7. E. Reyssat, Irrationalité de ζ(3), selon Apéry, Séminaire Delange--Pisot--Poitou, 20ème année (1978--1979), exposé no. 6


  8. R. Apéry, Interpolation de fractions continues et irrationalité de certaines constantes, Bulletin de la section des sciences du C.T.H.S III (1981), 37--53
  9. F. Beukers, Padé approximations in number theory, Lecture Notes in Math. 888, Springer-Verlag, Berlin (1981), 90--99
  10. F. Beukers, The values of polylogarithms, Topics in classical number theory, Colloq. Math. Soc. János Bolyai, Budapest (1981), 219--228


  11. L.A. Gutnik, On the irrationality of certain quantities involving ζ(3), Acta Arith. 42:3 (1983), 255--264


  12. J. Hancl, A simple proof of the irrationality of π4, Amer. Math. Monthly 93 (1986), 374--375


  13. F. Beukers, Irrationality proofs using modular forms, Astérisque 147--148 (1987), 271--283 (pdf)
  14. R. Dvornicich and C. Viola, Some remarks on Beukers' integrals, Colloq. Math. Soc. János Bolyai 51, North-Holland, Amsterdam (1987), 637--657
  15. E.A. Rukhadze, A lower bound for the approximation of ln2 by rational numbers, Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] no. 6 (1987), 25--29


  16. M. Hata, Legendre type polynomials and irrationality measures, J. Reine Angew. Math. 407:1 (1990), 99--125
  17. O.N. Vasilenko, Certain formulae for values of the Riemann zeta-function at integral points, Number theory and its applications, Proceedings of the science-theoretical conference (Tashkent, September 26--28, 1990), Tashkent (1990), 27 (pdf, dvi, gzip ps)


  18. M. Hata, Rational approximations to the dilogarithm, Trans. Amer. Math. Soc. 336:1 (1993), 363--387
  19. M. Hata, Rational approximations to π and some other numbers, Acta. Arith. 63 (1993), 335--349
  20. G. Rhin and C. Viola, On the irrationality measure of ζ(2), Ann. Inst. Fourier (Grenoble) 43:1 (1993), 85--109 (Numdam link)


  21. 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


  22. F. Beukers, Consequences of Apery's work on irrationality, Rencontres arithmétiques de Caen (1995), Preprint (pdf)
  23. M. Hata, A note on Beukers' integral, J. Austral. Math. Soc. Ser. A 58:2 (1995), 143--153


  24. Yu.V. Nesterenko, A few remarks on ζ(3), Mat. Zametki [Math. Notes] 59:6 (1996), 865--880 (gzip ps)
  25. M. Prévost, A new proof of the irrationality of ζ(3) using Padé approximants, J. Comput. Appl. Math. 67 (1996), 219--235 (pdf)
  26. G. Rhin and C. Viola, On a permutation group related to ζ(2), Acta Arith. 77:1 (1996), 23--56 (pdf)
  27. V.N. Sorokin, On the transcendence measure of the number π2, Mat. Sb. [Russian Acad. Sci. Sb. Math.] 187:12 (1996), 87--120


  28. M. Huttner, Équations différientielles fuchsiennes; Approximations du dilogarithme, de ζ(2) et ζ(3), Publ. IRMA, Lille (1997)
  29. C. Viola, Hypergeometric functions and irrationality measures, Analytic number theory (ed. Y. Motohashi), London Math. Soc. Lecture Note Ser. 247, Cambridge Univ. Press, Cambridge (1997), 353--360


  30. V.N. Sorokin, Apéry's theorem, Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] no. 3 (1998), 48--52


  31. M. Hata, A new irrationality measure for ζ(3), Acta Arith. 92:1 (2000), 47--57
  32. 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 (E-print math.NT/0008051)


  33. K. Ball and T. Rivoal, Irrationalité d'une infinité de valeurs de la fonction zêta aux entiers impairs, Invent. Math. 146:1 (2001), 193--207 (Springer link)
  34. L.A. Gutnik, On linear forms with coefficients in Nζ(1+N), Part 1, MPIM Preprint no. 15, Max-Plank-Institut fur Mathematik, Bonn (2001) (dvi, ps)
  35. L.A. Gutnik, On linear forms with coefficients in Nζ(1+N), Part 2, MPIM Preprint no. 104, Max-Plank-Institut fur Mathematik, Bonn (2001) (dvi, ps)
  36. G. Rhin and C. Viola, The group structure for ζ(3), Acta Arith. 97:3 (2001), 269--293 (pdf)
  37. T. Rivoal, Propriétés diophantiennes des valeurs de la fonction zêta de Riemann aux entiers impairs, Thèse de doctorat (29 juin 2001), Laboratoire SDAD, Université de Caen, Caen (link)
  38. 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)
  39. W. Zudilin, Irrationality of values of zeta function at odd integers, Uspekhi Mat. Nauk [Russian Math. Surveys] 56:2 (2001), 215--216 (pdf, gzip ps)
  40. W. Zudilin, Irrationality of values of zeta-function, Contemporary research in mathematics and mechanics, Proceedings of the XXIII Conference of Young Scientists of the Department of Mechanics and Mathematics (Moscow State University, April 9--14, 2001), Publ. Dept. Mech. Math. MSU, Moscow (2001), Part 2, 127--135 (E-print math.NT/0104249)
  41. W. 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 (pdf)
  42. W. 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 (pdf, gzip ps)


  43. T. Rivoal, Irrationalité d'au moins un des neuf nombres ζ(5), ζ(7), ..., ζ(21), Acta Arith. 103:2 (2002), 157--167 (E-print math.NT/0104221)
  44. 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 (pdf)
  45. 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 (pdf)
  46. W. Zudilin, A third-order Apéry-like recursion for ζ(5), Mat. Zametki [Math. Notes] 72:5 (2002), 796--800 (E-print math.NT/0206178)


  47. 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 (ps)
  48. T. Rivoal, Séries hypergéométriques et irrationalité des valeurs de la fonction zêta, Actes des journées arithmétiques de Lille (July, 2001), J. Théorie Nombres Bordeaux 15:1 (2003), 351--365 (ps)
  49. J. Sondow, Criteria for irrationality of Euler's constant, Proc. Amer. Math. Soc. 131:11 (2003), 3335--3344 (AMS link, E-print math.NT/0209070)
  50. C. Viola, Birational transformations and values of the Riemann zeta-function, Actes des 12èmes rencontres arithmétiques de Caen (June 29--30, 2001), J. Théorie Nombres Bordeaux 15:2 (2003), 561--592 (ps)
  51. 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 (ps, pdf)


  52. S. Fischler, Irrationalité de valeurs de zêta [d'après Apéry, Rivoal, ...], Séminaire Bourbaki 2002--2003, exposé numéro 910 (17 novembre 2002), Astérisque No. 294 (2004), 27--62 (E-print math.NT/0303066)
  53. W. Zudilin, Arithmetic of linear forms involving odd zeta values, J. Théorie Nombres Bordeaux 16:1 (2004), 251--291 (pdf, E-print math.NT/0206176)


  54. F. Calegari, Irrationality of certain p-adic periods for small p, Intern. Math. Research Notices 2005:20 (2005), 1235--1249 (Hindawi link, E-print math.NT/0408214)
  55. S. Zlobin, Expansion of multiple integrals in linear forms, Mat. Zametki 77:5 (2005), 683--706 (Russian pdf); English transl., Math. Notes 77:5--6 (2005), 630--652 (Springer link)
  56. S. Zlobin, Properties of coefficients of certain linear forms in generalized polylogarithms, Fundamental and Applied Mathematics 11:6 (2005), 41--58 (FPM link, pdf)


  57. M. Huttner, Constructible sets of linear differential equations and effective rational approximations of polylogarithmic functions, Israel J. Math. 153 (2006), 1--44
  58. C. Krattenthaler, T. Rivoal and W. Zudilin, Séries hypergéométriques basiques, q-analogues des valeurs de la fonction zêta et formes modulaires, J. Inst. Math. Jussieu 5:1 (2006), 53--79 (CUP link, E-print math.NT/0311033)


  59. C. Krattenthaler and T. Rivoal, Hypergéométrie et fonction zêta de Riemann, Mem. Amer. Math. Soc. 186 (2007), no. 875, x+87 pp. (E-print math.NT/0311114)
  60. C. Krattenthaler and T. Rivoal, An identity of Andrews, multiple integrals, and very-well-poised hypergeometric series, Ramanujan J. 13:1-3 (2007), 203--219 (Springer link, E-print math.CA/0312148)


  61. F. Beukers, Irrationality of some p-adic L-values, Acta Math. Sin. (Engl. Ser.) 24:4 (2008), 663--686 (Springer link, E-print math.NT/0603277)


  62. S. Fischler and W. Zudilin, A refinement of Nesterenko's linear independence criterion with applications to zeta values, Preprint MPIM 2009-35 (May 2009), 24 pages

Compiled by

Wadim Zudilin / w [dot] zudilin [at] math [dot] ru [dot] nl ; wzudilin [at] gmail [dot] com

Institute for Mathematics, Astrophysics and Particle Physics
Radboud Universiteit, PO Box 9010
6500 GL Nijmegen, The Netherlands

Started on 5 February 2002
The collection was last modified on 19 September 2009