Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
E. Ghate, On monomial relations between Eisenstein series. J. Ramanujan Math. Soc. 15(2), 71–79 (2000)
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
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)
D. Lanphier, Maass operators and van der Pol-type identities for Ramanujan’s tau function. Acta Arith. 113, 157–167 (2004)
H. Maass, Siegel Modular Forms and Dirichlet Series. Lecture Notes in Mathematics, vol. 216 (Springer, Berlin, 1971)
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
D. Niebur, A formula for Ramanujan’s τ-function. Ill. J. Math. 19, 448–449 (1975)
B. Ramakrishnan, B. Sahu, Rankin–Cohen brackets and van der Pol-type identities for the Ramanujan τ-function. Preprint (2007)
R.A. Rankin, The construction of automorphic forms from the derivatives of a given form. J. Indian Math. Soc. 20, 103–116 (1956)
G. Shimura, The special values of the zeta functions associated with cusp forms. Commun. Pure Appl. Math. 29, 783–804 (1976)
G. Shimura, Differential operators, holomorphic projections, and singular forms. Duke Math. J. 76, 141–173 (1994)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer India
About this chapter
Cite this chapter
Murty, M.R., Murty, V.K. (2013). Some Nonlinear Identities for Divisor Functions. In: The Mathematical Legacy of Srinivasa Ramanujan. Springer, India. https://doi.org/10.1007/978-81-322-0770-2_8
Download citation
DOI: https://doi.org/10.1007/978-81-322-0770-2_8
Published:
Publisher Name: Springer, India
Print ISBN: 978-81-322-0769-6
Online ISBN: 978-81-322-0770-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)