Abstract
Ramanujan (1916) and Shen (1999) discovered differential equations for classical Eisenstein series. Motivated by them, we derive new differential equations for Eisenstein series of level 2 from the second kind of Jacobi theta function. This gives a new characterization of a system of differential equations by Ablowitz–Chakravarty–Hahn (2006), Hahn (2008), Kaneko–Koike (2003), Maier (2011), Nidelan (2022) and Toh (2011). As application, we show some arithmetic results on Ramanujan’s tau function.
Similar content being viewed by others
Data Availability
All data generated during this study are included in this article. We have no conflicts of interest to disclose.
References
Ablowitz, M.J., Chakravarty, S., Hahn, H.: Integrable systems and modular forms of level 2. J. Phys. A 39(50), 15341–15353 (2006)
Balakrishnan, J.S., Ono, K., Tsai, W.L.: Even values of Ramanujan’s tau-function. Matematica 1(2), 395–403 (2022)
Bennett, M.A., Gherga, A., Patel, V., Siksek, S.: Odd values of the Ramanujan tau function. Math. Ann. 382(1–2), 203–238 (2022)
Brundaban, S., Ramakrishnan, B.: Identities for the Ramanujan Tau Function and Certain Convolution Sum Identities for the Divisor Functions. Ramanujan Math. Soc. Lect. Notes Ser., vol. 23, pp. 63–75. Ramanujan Math. Soc., Mysore (2016)
Chan, H.H., Chua, K.S.: Representations of integers as sums of 32 squares. Ramanujan J. Rankin memorial issues 7, 79–89 (2003)
Deligne, P.: La conjecture de Weil I. Inst. Hautes Etudes Sci. Publ. Math. 43, 273–307 (1974)
Hahn, H.: convolution sums of some functions on divisors. Rocky Mt. J. Math. 37(5), 1593–1622 (2007)
Hahn, H.: Eisenstein series associated with \(\Gamma _0(2)\). Ramanujan J. 15(2), 235–257 (2008)
Huard, J.G., Ou, Z.M., Spearman, B.K., Williams, K.S.: Elementary evaluation of certain convolution sums involving divisor functions. Number Theory Millenn. II, 229–274 (2002)
Imamoḡlu, O., Kohnen, W.: Representations of integers as sums of an even number of squares. Math. Ann. 333(4), 815–829 (2005)
Kaneko, M., Koike, M.: On modular forms arising from a differential equation of hypergeometric type. Ramanujan J. 7(1–3), 145–164 (2003)
Lehmer, D.H.: The vanishing of Ramanujan’s \(\tau (n)\). Duke Math. J. 1, 429–433 (1947)
Lakein, K., Larsen, A.: Some remarks on small values of \(\tau (n)\). Arch. Math. 117(6), 635–645 (2021)
Maier, R.S.: Nonlinear differential equations satisfied by certain classical modular forms. Manuscr. Math. 134(1–2), 1–42 (2011)
Milne, S.C.: Hankel determinants of Eisenstein series, Symbolic computation, number theory, special functions, physics and combinatorics. Gainesville, FL (1999). Dev. Math., 4, Kluwer Acad. Publ., Dordrecht, 2001, pp. 171–188
Milne, S.C.: New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Natl. Acad. Sci. U.S.A. 93(26), 15004–15008 (1996)
Mordell, L.J.: On Mr. Ramanujan’s empirical expansions of modular functions. Proc. Camb. Philos. Soc. 19, 117–124 (1917)
Moree, P.: On some claims in Ramanujan’s ‘unpublished’ manuscript on the partition and tau functions. Ramanujan J. 8(3), 317–330 (2004)
Murty, M.R., Murty, V.K., Shorey, T.N.: Odd values of the Ramanujan \(\tau \)-function. Bull. Soc. Math. Fr. 115(3), 391–395 (1987)
Nikdelan, Y.: Ramanujan-type systems of nonlinear ODEs for \(\Gamma _{0}(2)\) and \(\Gamma _{0}(3)\). Exp. Math. 40, 409–431 (2022)
Ramanujan, S.: On certain arithmetical functions. Trans. Camb. Philos. Soc. 22, 159–184 (1916)
Shen, L.C.: On the logarithmic derivative of a theta function and a fundamental identity of Ramanujan. J. Math. Anal. Appl. 177(1), 299–307 (1993)
Toh, P.C.: Differential equations satisfied by Eisenstein series of level 2. Ramanujan J. 25(2), 179–194 (2011)
Acknowledgements
This work is based on the author’s talk in Hiroshima-Sendai Number Theory Workshop at Hiroshima University, Japan, in July 2023. He thanks its organizers and the audience who gave valuable comments and suggestions. He is also grateful for the anonymous referees for careful reading and helpful comments to improve the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kobayashi, M. Ramanujan–Shen’s differential equations for Eisenstein series of level 2. Res. number theory 10, 41 (2024). https://doi.org/10.1007/s40993-024-00527-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40993-024-00527-4
Keywords
- Divisor sum
- Eisenstein series
- Jacobi theta functions
- Modular forms
- Ramanujan’s differential equations
- Ramanujan’s tau function
- Serre derivative
- Sum of squares