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Talk: On the ubiquity of modular forms and Apéry-like numbers (Tulane)

On the ubiquity of modular forms and Apéry-like numbers (Tulane)

Date: 2013/10/17
Occasion: Algebra and Combinatorics Seminar
Place: Tulane University

Abstract

In the first part of this talk, we give examples from the theories of short random walks, binomial congruences, positivity of rational functions and series for $$1/\pi$$, in which modular forms and Apéry-like numbers appear naturally (though not necessarily obviously). Each example is taken from personal research of the speaker. The second part, which is based on joint work with Bruce C. Berndt, is motivated by the secant Dirichlet series $$\psi_s(\tau) = \sum_{n = 1}^{\infty} \frac{\sec(\pi n \tau)}{n^s}$$, recently introduced and studied by Lalín, 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 of general Eisenstein series.