Special values of trigonometric Dirichlet series and Eichler integrals

TL;DR

This paper develops a general method to evaluate trigonometric Dirichlet series involving elementary functions, quadratic irrationals, and integer exponents, unifying many classical results through Eichler integrals and polylogarithms.

Contribution

It introduces a comprehensive theorem that evaluates these series for various functions and parameters, connecting them to Eichler integrals and polylogarithms, and unifies previous isolated results.

Findings

Unified evaluation formula for trigonometric Dirichlet series.

Connection between series and derivatives of Eichler integrals.

Generalization of classical results by Lerch, Ramanujan, and Berndt.

Abstract

We provide a general theorem for evaluating trigonometric Dirichlet series of the form $\sum_{n \geq 1} \frac{f (\pi n \tau)}{n^s}$, where $f$ is an arbitrary product of the elementary trigonometric functions, $\tau$ a real quadratic irrationality and $s$ an integer of the appropriate parity. This unifies a number of evaluations considered by many authors, including Lerch, Ramanujan and Berndt. Our approach is based on relating the series to combinations of derivatives of Eichler integrals and polylogarithms.