The Ramanujan Journal

, Volume 29, Issue 1–3, pp 135–144 | Cite as

Complex series for 1/π

  • Heng Huat Chan
  • James Wan
  • Wadim Zudilin


Many series for 1/π were discovered since the appearance of S. Ramanujan’s famous paper “Modular equations and approximation to π” published in 1914. Almost all these series involve only real numbers. Recently, in an attempt to prove a series for 1/π discovered by Z.-W. Sun, the authors found that a series for 1/π involving complex numbers is needed. In this article, we illustrate a method that would allow us to prove series of this type.


Hypergeometric series Singular moduli Lambert series 

Mathematics Subject Classification (2000)

11F11 11F03 11Y60 33C05 33C20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory, 2nd edn. Graduate Text in Math, vol. 41. Springer, New York (1990) zbMATHCrossRefGoogle Scholar
  2. 2.
    Bauer, G.: Von den Coeffizienten der Reihen von Kugelfunctionen einer Variablen. J. Reine Angew. Math. 56, 101–121 (1859) zbMATHCrossRefGoogle Scholar
  3. 3.
    Berndt, B.C., Chan, H.H., Liaw, W.-C.: On Ramanujan’s quartic theory of elliptic functions. J. Number Theory 88(1), 129–156 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Borwein, J.M., Borwein, P.B.: Pi and the AGM; A Study in Analytic Number Theory and Computational Complexity. Wiley, New York (1987) zbMATHGoogle Scholar
  5. 5.
    Chan, H.H., Zudilin, W.: New representations for Apéry-like sequences. Mathematika 56(1), 107–117 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Chan, H.H., Chan, S.H., Liu, Z.G.: Domb’s numbers and Ramanujan–Sato type series for 1/π. Adv. Math. 186, 396–410 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chan, H.H., Wan, J., Zudilin, W.: Legendre polynomials and Ramanujan-type series for 1/π. Isr. J. Math. (to appear) Google Scholar
  8. 8.
    Chudnovsky, D.V., Chudnovsky, G.V.: Approximations and complex multiplication according to Ramanujan. In: Ramanujan Revisited, Urbana-Champaign, IL, 1987, pp. 375–472. Academic Press, Boston (1988) Google Scholar
  9. 9.
    Guillera, J., Zudilin, W.: “Divergent” Ramanujan-type supercongruences. Proc. Am. Math. Soc. 140, 765–777 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ramanujan, S.: Modular equations and approximations to π. Q. J. Math. 45, 350–372 (1914) Google Scholar
  11. 11.
    Sato, T.: Apéry numbers and Ramanujan’s series for 1/π. Abstract of a talk presented at the annual meeting of the Mathematical Society of Japan (28–31 March 2002) Google Scholar
  12. 12.
    Sun, Z.-W.: List of conjectural series for powers of π and other constants. Preprint arXiv:1102.5649 [math.CA] (2011)

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsNational University of SingaporeSingaporeSingapore
  2. 2.School of Mathematical and Physical SciencesThe University of NewcastleCallaghanAustralia

Personalised recommendations