Algebra and Topology Seminar

4pm Tuesday 11 October 2005
Bruce Berndt
University of Illinois

Ramanujan's contributions to Eisenstein series especially in his Lost Notebook

Eisenstein series are the building blocks of modular forms; in particular, every analytic modular form on the full modular group can be represented as a polynomial in two particular Eisenstein series. For Ramanujan, the primary Eisenstein series were, in his notation, P(q), Q(q), and R(q). We provide a survey of many of Ramanujan's discoveries about Eisenstein series; most of the theorems are found in his lost notebook. 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 arising from Eisenstein series.




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