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Talk: Trigonometric Dirichlet series and Eichler integrals (Dalhousie)

Trigonometric Dirichlet series and Eichler integrals (Dalhousie)

Date: 2014/10/20
Occasion: Number Theory and Experimental Mathematics Day
Place: Dalhousie University

Abstract

This talk 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 Eisenstein series of higher level. 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 by Conrey, Farmer and Imamoglu as well as El-Guindy and Raji on zeros of period polynomials of Hecke eigenforms in the case of level \(1\). This talk is based on joint work with Bruce C. Berndt.

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