*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 Ap\'ery-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 Lal\'{\i}n,
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. |