Some Nonlinear Identities for Divisor Functions

  • M. Ram Murty
  • V. Kumar Murty


In his 1916 paper, Ramanujan derived a series of nonlinear identities for a class of divisor functions. We now know that these identities arise as a consequence of the low dimension of certain spaces of modular forms. In this chapter, we explain these identities as well as indicate the relation to the theory of quasi-modular forms.


  1. 59.
    E. Ghate, On monomial relations between Eisenstein series. J. Ramanujan Math. Soc. 15(2), 71–79 (2000) MathSciNetzbMATHGoogle Scholar
  2. 90.
    M. Kaneko, D. Zagier, A generalized Jacobi theta function and quasimodular forms, in The Moduli Space of Curves. Progress in Mathematics, vol. 129 (Birkhäuser, Boston, 1995), pp. 165–172 CrossRefGoogle Scholar
  3. 104.
    D.B. Lahiri, On Ramanujan’s function τ(n) and the divisor function σ k(n), I. Bull. Calcutta Math. Soc. 38, 193–206 (1946). II, 39, 33–52 (1947) MathSciNetzbMATHGoogle Scholar
  4. 109.
    D. Lanphier, Maass operators and van der Pol-type identities for Ramanujan’s tau function. Acta Arith. 113, 157–167 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 118.
    H. Maass, Siegel Modular Forms and Dirichlet Series. Lecture Notes in Mathematics, vol. 216 (Springer, Berlin, 1971) zbMATHGoogle Scholar
  6. 121.
    F. Martin, E. Royer, Formes modulaires et périodes, in Formes modulaires et transcendance, Sémin. Congr., vol. 12 (Soc. Math. France, Paris, 2005), pp. 1–17 Google Scholar
  7. 149.
    D. Niebur, A formula for Ramanujan’s τ-function. Ill. J. Math. 19, 448–449 (1975) MathSciNetzbMATHGoogle Scholar
  8. 167.
    B. Ramakrishnan, B. Sahu, Rankin–Cohen brackets and van der Pol-type identities for the Ramanujan τ-function. Preprint (2007) Google Scholar
  9. 170.
    R.A. Rankin, The construction of automorphic forms from the derivatives of a given form. J. Indian Math. Soc. 20, 103–116 (1956) MathSciNetzbMATHGoogle Scholar
  10. 189.
    G. Shimura, The special values of the zeta functions associated with cusp forms. Commun. Pure Appl. Math. 29, 783–804 (1976) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 190.
    G. Shimura, Differential operators, holomorphic projections, and singular forms. Duke Math. J. 76, 141–173 (1994) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer India 2013

Authors and Affiliations

  • M. Ram Murty
    • 1
  • V. Kumar Murty
    • 2
  1. 1.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations