On the ubiquity of modular forms and Apery-like numbers
Speaker: Dr. Armin Straub (MPIM and UIUC)
Time: 4:00pm
Date: Monday 18th November 2013
Location: CASL Seminar Room, Block 8, Belfield Office Park
Abstract:
Apery-like numbers are special sequences which are modelled after and share many of the properties of the numbers that underlie Apery's proof of the irrationality of \zeta(3). In the course of several examples, we demonstrate how these numbers and their connection with modular forms feature in various, apparently unrelated, problems. The examples are taken from personal research of the speaker and include the theories of short random walks, binomial congruences, positivity of rational functions and series for 1/\pi.
Time permitting, we briefly report on joint work with Bruce C. Berndt on a trigonometric Dirichlet series which was recently introduced and studied by Lalin, Rodrigue and Rogers as a variation of results of Ramanujan. We review some of its properties, prove a conjecture on special values of this Dirichlet series, and put these into the context of Eichler integrals ofgeneral Eisenstein series.
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