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Ramanujan’s Contributions to Eisenstein Series, Especially in His Lost Notebook

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Book cover Number Theoretic Methods

Part of the book series: Developments in Mathematics ((DEVM,volume 8))

Abstract

The primary Eisenstein series considered in this paper are, in Ramanujan’s notation, P(q), Q(q), and R(q). In more standard notation, Q(q) = E 4(τ) and R(q) = E 6(τ), where q = exp(2πiτ). This paper provides a survey of many of Ramanujan’s discoveries about Eisenstein series; most of the theorems are found in his lost notebook and are due originally to Ramanujan. Some of the topics examined are formulas for the power series coefficients of certain quotients of Eisenstein series, the role of Eisenstein series in proving congruences for the partition function p(n), representations of Eisenstein series as sums of quotients of Dedekind eta-functions, a family of infinite series represented by polynomials in P, Q, and R, and approximations and exact formulas for π.

Research partially supported by grant MDA904-00-1-0015 from the National Security Agency.

Research partially supported by a grant from the Number Theory Foundation.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL, 61801, USA

    Bruce C. Berndt & Ae Ja Yee

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  1. Bruce C. Berndt
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  2. Ae Ja Yee
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Editors and Affiliations

  1. Graduate School of Advanced Technology, University of Kinki, Iizuka, Japan

    Shigeru Kanemitsu

  2. Institute of Mathematics, Academia Sinica, Beijing, China

    Chaohua Jia

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Berndt, B.C., Yee, A.J. (2002). Ramanujan’s Contributions to Eisenstein Series, Especially in His Lost Notebook. In: Kanemitsu, S., Jia, C. (eds) Number Theoretic Methods. Developments in Mathematics, vol 8. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3675-5_3

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  • DOI: https://doi.org/10.1007/978-1-4757-3675-5_3

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-5239-4

  • Online ISBN: 978-1-4757-3675-5

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