Series: Mathematics Colloquium

Date: Thursday, March 6, 2003

Time: 4:30 - 5:30 PM

Place: 102 McAllister Building

Host: George Andrews

Refreshments: 4:00 - 4:30 PM, in 212 McAllister

Speaker: Bruce Berndt, University of Illinois

Title: Ramanujan's Contributions to Eisenstein Series, Especially in
his Lost Notebook

Abstract:

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 pi arising from Eisenstein series.