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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

Abstract

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.

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