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Special values of trigonometric Dirichlet series and Eichler integrals

Special values of trigonometric Dirichlet series and Eichler integrals
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.

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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},
}