Talk: Special values of trigonometric Dirichlet series (NIST)

Special values of trigonometric Dirichlet series (NIST)

Date: 2015/06/03
Occasion: 13th International Symposium on Orthogonal Polynomials, Special Functions and Applications, Minisymposium on the Legacy of Ramanujan
Place: NIST


In his first letter to Hardy and in several entries of his notebooks, Ramanujan recorded many evaluations of trigonometric Dirichlet series such as \[ \sum_{n=1}^\infty \frac{\coth(\pi n)}{n^{2r-1}} = \frac12 (2\pi)^{2r-1} \sum_{m=0}^r (-1)^{m+1} \frac{B_{2m}}{(2m)!} \frac{B_{2(r-m)}}{(2(r-m))!}, \] which, in fact, goes back to Cauchy and Lerch. A recent example is the secant Dirichlet series \(\psi_s(\tau) = \sum_{n=1}^\infty \frac{\sec(\pi n \tau)}{n^s}\), for which Lalín, Rodrigue and Rogers conjecture, and partially prove, that its values \(\psi_{2 m}(\sqrt{r})\), with \(r > 0\) rational, are rational multiples of \(\pi^{2m}\). We give an overview of special values of such Dirichlet series and their connection with the theory of modular forms.

This talk includes joint work with Bruce C. Berndt.


1.70 MB Slides (PDF, 70 pages) 1232