Ramanujan’s Formula for ζ(2n + 1)

Chapter

Abstract

Ramanujan made many beautiful and elegant discoveries in his short life of 32 years, and one of them that has attracted the attention of several mathematicians over the years is his intriguing formula for ζ(2n + 1). To be sure, Ramanujan’s formula does not possess the elegance of Euler’s formula for ζ(2n), nor does it provide direct arithmetical information. But, one of the goals of this survey is to convince readers that it is indeed a remarkable formula. In particular, we discuss the history of Ramanujan’s formula, its connection to modular forms, as well as the remarkable properties of the associated polynomials. We also indicate analogues, generalizations and opportunities for further research.

References

  1. 1.
    G.E. Andrews, B.C. Berndt, Ramanujan’s Lost Notebook, Part IV (Springer, New York, 2013)CrossRefMATHGoogle Scholar
  2. 2.
    R. Apéry, Interpolation de fractions continues et irrationalite de certaines constantes. Bull. Section des Sci., Tome III (Bibliothéque Nationale, Paris, 1981), pp. 37–63MATHGoogle Scholar
  3. 3.
    T.M. Apostol, Generalized Dedekind sums and transformation formulae of certain Lambert series. Duke Math. J. 17, 147–157 (1950)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    T.M. Apostol, Letter to Emil Grosswald, January 24, 1973Google Scholar
  5. 5.
    B.C. Berndt, Generalized Dedekind eta-functions and generalized Dedekind sums. Trans. Am. Math. Soc. 178, 495–508 (1973)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    B.C. Berndt, Ramanujan’s formula for ζ(2n + 1), in Professor Srinivasa Ramanujan Commemoration Volume (Jupiter Press, Madras, 1974), pp. 2–9Google Scholar
  7. 7.
    B.C. Berndt, Dedekind sums and a paper of G.H. Hardy. J. Lond. Math. Soc. (2) 13, 129–137 (1976)Google Scholar
  8. 8.
    B.C. Berndt, Modular transformations and generalizations of several formulae of Ramanujan. Rocky Mt. J. Math. 7, 147–189 (1977)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    B.C. Berndt, Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan. J. Reine Angew. Math. 304, 332–365 (1978)MathSciNetMATHGoogle Scholar
  10. 10.
    B.C. Berndt, Ramanujan’s Notebooks, Part II (Springer, New York, 1989)MATHGoogle Scholar
  11. 11.
    B.C. Berndt, Ramanujan’s Notebooks, Part III (Springer, New York, 1991)CrossRefMATHGoogle Scholar
  12. 12.
    B.C. Berndt, An unpublished manuscript of Ramanujan on infinite series identities. J. Ramanujan Math. Soc. 19, 57–74 (2004)MathSciNetMATHGoogle Scholar
  13. 13.
    B.C. Berndt, A. Straub, On a secant Dirichlet series and Eichler integrals of Eisenstein series. Math. Z. 284(3), 827–852 (2016). doi:10.1007/s00209-016-1675-0 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    R. Bodendiek, Über verschiedene Methoden zur Bestimmung der Transformationsformeln der achten Wurzeln der Integralmoduln k 2(τ) und k 2(τ), ihrer Logarithmen sowie gewisser Lambertscher Reihen bei beliebigen Modulsubstitutionen, Dissertation der Universität Köln, 1968Google Scholar
  15. 15.
    R. Bodendiek, U. Halbritter, Über die Transformationsformel von logη(τ) und gewisser Lambertscher Reihen. Abh. Math. Semin. Univ. Hambg. 38, 147–167 (1972)CrossRefMATHGoogle Scholar
  16. 16.
    G. Bol, Invarianten linearer Differentialgleichungen. Abh. Math. Semin. Univ. Hambg. 16, 1–28 (1949)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    D.M. Bradley, Series acceleration formulas for Dirichlet series with periodic coefficients. Ramanujan J. 6, 331–346 (2002)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    K. Chandrasekharan, R. Narasimhan, Hecke’s functional equation and arithmetical identities. Ann. Math. 74, 1–23 (1961)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    J.B. Conrey, D.W. Farmer, Ö. Imamoglu, The nontrivial zeros of period polynomials of modular forms lie on the unit circle. Int. Math. Res. Not. 2013(20), 4758–4771 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    A. El-Guindy, W. Raji, Unimodularity of roots of period polynomials of Hecke eigenforms. Bull. Lond. Math. Soc. 46(3), 528–536 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    B. Ghusayni, The value of the zeta function at an odd argument. Int. J. Math. Comp. Sci. 4, 21–30 (2009)MathSciNetMATHGoogle Scholar
  22. 22.
    H.-J. Glaeske, Eine einheitliche Herleitung einer gewissen Klasse von Transformationsformeln der analytischen Zahlentheorie (I), Acta Arith. 20, 133–145 (1972)MathSciNetMATHGoogle Scholar
  23. 23.
    H.-J. Glaeske, Eine einheitliche Herleitung einer gewissen Klasse von Transformationsformeln der analytischen Zahlentheorie (II). Acta Arith. 20, 253–265 (1972)MathSciNetMATHGoogle Scholar
  24. 24.
    J.W.L. Glaisher, On the series which represent the twelve elliptic and the four zeta functions. Mess. Math. 18, 1–84 (1889)MathSciNetGoogle Scholar
  25. 25.
    E. Grosswald, Die Werte der Riemannschen Zeta-funktion an ungeraden Argumentstellen. Nachr. Akad. Wiss. Göttingen 9–13 (1970)Google Scholar
  26. 26.
    E. Grosswald, Remarks concerning the values of the Riemann zeta function at integral, odd arguments. J. Number Theor. 4(3), 225–235 (1972)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    E. Grosswald, Comments on some formulae of Ramanujan. Acta Arith. 21, 25–34 (1972)MathSciNetMATHGoogle Scholar
  28. 28.
    A.P. Guinand, Functional equations and self-reciprocal functions connected with Lambert series. Quart. J. Math. 15, 11–23 (1944)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    A.P. Guinand, Some rapidly convergent series for the Riemann ξ-function. Quart. J. Math. Ser. (2) 6, 156–160 (1955)Google Scholar
  30. 30.
    S. Gun, M.R. Murty, P. Rath, Transcendental values of certain Eichler integrals. Bull. Lond. Math. Soc. 43, 939–952 (2011)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    S. Iseki, The transformation formula for the Dedekind modular function and related functional equations. Duke Math. J. 24, 653–662 (1957)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    S. Jin, W. Ma, K. Ono, K. Soundararajan, The Riemann Hypothesis for period polynomials of modular forms. Proc. Natl. Acad. Sci. USA. 113(10), 2603–2608 (2016). doi:10.1073/pnas.1600569113 MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    S. Kanemitsu, T. Kuzumaki, Transformation formulas for Lambert series. Šiauliai Math. Semin. 4(12), 105–123 (2009)MathSciNetMATHGoogle Scholar
  34. 34.
    S. Kanemitsu, Y. Tanigawa, M. Yoshimoto, On rapidly convergent series for the Riemann zeta-values via the modular relation. Abh. Math. Semin. Univ. Hambg. 72, 187–206 (2002)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    S. Kanemitsu, Y. Tanigawa, M. Yoshimoto, Ramanujan’s formula and modular forms, in Number Theoretic Methods - Future Trends, ed. by S. Kanemitsu, C. Jia (Kluwer, Dordrecht, 2002), pp. 159–212CrossRefGoogle Scholar
  36. 36.
    K. Katayama, On Ramanujan’s formula for values of Riemann zeta-function at positive odd integers. Acta Arith. 22, 149–155 (1973)MathSciNetMATHGoogle Scholar
  37. 37.
    K. Katayama, Zeta-functions, Lambert series and arithmetic functions analogous to Ramanujan’s τ-function. II. J. Reine Angew. Math. 282, 11–34 (1976)MathSciNetMATHGoogle Scholar
  38. 38.
    M. Katsurada, Asymptotic expansions of certain q-series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107, 269–298 (2003)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    M. Katsurada, Complete asymptotic expansions for certain multiple q-integrals and q-differentials of Thomae-Jackson type. Acta Arith. 152, 109–136 (2012)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Y. Komori, K. Matsumoto, H. Tsumura, Barnes multiple zeta-function, Ramanujan’s formula, and relevant series involving hyperbolic functions. J. Ramanujan Math. Soc. 28(1), 49–69 (2013)MathSciNetMATHGoogle Scholar
  41. 41.
    S. Kongsiriwong, A generalization of Siegel’s method. Ramanujan J. 20(1), 1–24 (2009)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    M.N. Lalín, M.D. Rogers, Variations of the Ramanujan polynomials and remarks on ζ(2j + 1)∕π 2j+1. Funct. Approx. Comment. Math. 48(1), 91–111 (2013)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    M.N. Lalín, C.J. Smyth, Unimodularity of roots of self-inversive polynomials. Acta Math. Hungar. 138, 85–101 (2013)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    M. Lerch, Sur la fonction ζ(s) pour valeurs impaires de l’argument. J. Sci. Math. Astron. pub. pelo Dr. F. Gomes Teixeira, Coimbra 14, 65–69 (1901)Google Scholar
  45. 45.
    J. Lewis, D. Zagier, Period functions for Maass wave forms. I. Ann. Math. 153, 191–258 (2001)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    S.-G. Lim, Generalized Eisenstein series and several modular transformation formulae. Ramanujan J. 19(2), 121–136 (2009)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    S.-G. Lim, Infinite series identities from modular transformation formulas that stem from generalized Eisenstein series. Acta Arith. 141(3), 241–273 (2010)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    S.-G. Lim, Character analogues of infinite series from a certain modular transformation formula. J. Korean Math. Soc. 48(1), 169–178 (2011)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    S.L. Malurkar, On the application of Herr Mellin’s integrals to some series. J. Indian Math. Soc. 16, 130–138 (1925/1926)Google Scholar
  50. 50.
    M. Mikolás, Über gewisse Lambertsche Reihen, I: Verallgemeinerung der Modulfunktion η(τ) und ihrer Dedekindschen Transformationsformel. Math. Z. 68, 100–110 (1957)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    M.R. Murty, C. Smyth, R.J. Wang, Zeros of Ramanujan polynomials. J. Ramanujan Math. Soc. 26, 107–125 (2011)MathSciNetMATHGoogle Scholar
  52. 52.
    T.S. Nanjundiah, Certain summations due to Ramanujan, and their generalisations. Proc. Indian Acad. Sci. Sect. A 34, 215–228 (1951)MathSciNetMATHGoogle Scholar
  53. 53.
    P. Panzone, L. Piovan, M. Ferrari, A generalization of Iseki’s formula. Glas. Mat. Ser. III (66) 46(1), 15–24 (2011)Google Scholar
  54. 54.
    V. Paşol, A.A. Popa, Modular forms and period polynomials. Proc. Lond. Math. Soc. 107(4), 713–743 (2013)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    S. Ramanujan, Some formulae in the analytic theory of numbers. Mess. Math. 45, 81–84 (1916)MathSciNetGoogle Scholar
  56. 56.
    S. Ramanujan, On certain trigonometric sums and their applications in the theory of numbers. Trans. Cambridge Philos. Soc. 22, 259–276 (1918)Google Scholar
  57. 57.
    S. Ramanujan, Collected Papers (Cambridge University Press, Cambridge, 1927); reprinted by Chelsea, New York, 1962; reprinted by the American Mathematical Society, Providence, RI, 2000Google Scholar
  58. 58.
    S. Ramanujan, Notebooks, 2 vols. (Tata Institute of Fundamental Research, Bombay, 1957; 2nd ed., 2012)Google Scholar
  59. 59.
    S. Ramanujan, The Lost Notebook and Other Unpublished Papers (Narosa, New Delhi, 1988)MATHGoogle Scholar
  60. 60.
    M.B. Rao, M.V. Ayyar, On some infinite series and products. Part I. J. Indian Math. Soc. 15, 150–162 (1923/1924)Google Scholar
  61. 61.
    S.N. Rao, A proof of a generalized Ramanujan identity. J. Mysore Univ. Sect. B 28, 152–153 (1981–1982)Google Scholar
  62. 62.
    M.J. Razar, Values of Dirichlet series at integers in the critical strip, in Modular Functions of One Variable VI. Lecture Notes in Mathematics, vol. 627, ed. by J.-P. Serre, D.B. Zagier (Springer, Berlin/Heidelberg, 1977), pp. 1–10Google Scholar
  63. 63.
    H. Riesel, Some series related to infinite series given by Ramanujan. BIT 13, 97–113 (1973)MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    H.F. Sandham, Some infinite series. Proc. Am. Math. Soc. 5, 430–436 (1954)MathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    F.P. Sayer, The sum of certain series containing hyperbolic functions. Fibonacci Quart. 14, 215–223 (1976)MathSciNetMATHGoogle Scholar
  66. 66.
    O. Schlömilch, Ueber einige unendliche Reihen. Ber. Verh. K. Sachs. Gesell. Wiss. Leipzig 29, 101–105 (1877)MATHGoogle Scholar
  67. 67.
    O. Schlömilch, Compendium der höheren Analysis. zweiter Band, 4th ed. (Friedrich Vieweg und Sohn, Braunschweig, 1895)Google Scholar
  68. 68.
    C.L. Siegel, A simple proof of \(\eta (-1/\tau ) =\eta (\tau )\sqrt{\tau /i}\). Mathematika 1, 4 (1954)MathSciNetCrossRefMATHGoogle Scholar
  69. 69.
    R. Sitaramachandrarao, Ramanujan’s formula for ζ(2n + 1), Madurai Kamaraj University Technical Report 4, pp. 70–117Google Scholar
  70. 70.
    J.R. Smart, On the values of the Epstein zeta function. Glasgow Math. J. 14, 1–12 (1973)MathSciNetCrossRefMATHGoogle Scholar
  71. 71.
    L. Veps̆tas, On Plouffe’s Ramanujan identities. Ramanujan J. 27, 387–408 (2012)Google Scholar
  72. 72.
    G.N. Watson, Theorems stated by Ramanujan II. J. Lond. Math. Soc. 3, 216–225 (1928)CrossRefMATHGoogle Scholar
  73. 73.
    A. Weil, Remarks on Hecke’s lemma and its use, in Algebraic Number Theory: Papers Contributed for the Kyoto International Symposium, 1976, ed. by S. Iyanaga (Japan Society for the Promotion of Science, Tokyo, 1977), pp. 267–274Google Scholar
  74. 74.
    D.B. Zagier, Periods of modular forms and Jacobi theta functions. Invent. Math. 104, 449–465 (1991)MathSciNetCrossRefMATHGoogle Scholar
  75. 75.
    N. Zhang, Ramanujan’s formula and the values of the Riemann zeta-function at odd positive integers (Chinese). Adv. Math. Beijing 12, 61–71 (1983)MathSciNetMATHGoogle Scholar
  76. 76.
    N. Zhang, S. Zhang, Riemann zeta function, analytic functions of one complex variable. Contemp. Math. 48, 235–241 (1985)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana–ChampaignUrbanaUSA
  2. 2.Department of Mathematics and StatisticsUniversity of South AlabamaMobileUSA

Personalised recommendations