Armin Straub — The Ramanujan Journal — Special issue dedicated to Marvin Knopp — Volume 41, Number 1, 2016, Pages 269-285
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.Download
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| trigonometricdirichletseries.pdf | 370.42 KB | Preprint (PDF, 17 pages) | 3072 |
BibTeX
@article{trigonometricdirichletseries-2016,
author = {Armin Straub},
title = {Special values of trigonometric {D}irichlet series and {E}ichler integrals},
journal = {The Ramanujan Journal},
year = {2016},
volume = {41},
number = {1},
pages = {269--285},
doi = {10.1007/s11139-015-9698-4},
}