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

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

Date: 2013/10/09
Occasion: Algorithmic Combinatorics Seminar
Place: RISC, Johannes Kepler University, Linz


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, which include a modular functional equation when \(s\) is even, and demonstrate that the values \(\psi_{2 m}(\sqrt{r})\), with \(r > 0\) rational, are rational multiples of \(\pi^{2 m}\). These properties are then put into the context of Eichler integrals of general Eisenstein series. In particular, we determine the period polynomials of such Eichler integrals and indicate that they appear to give rise to unimodular polynomials, an observation which complements recent results on zeros of period polynomials of cusp forms by Conrey, Farmer and Imamoglu.


6.08 MB Slides (PDF, 93 pages) 1606