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The Ramanujan Journal

, Volume 41, Issue 1–3, pp 269–285 | Cite as

Special values of trigonometric Dirichlet series and Eichler integrals

  • Armin Straub
Article
  • 204 Downloads

Abstract

We provide a general theorem for evaluating trigonometric Dirichlet series of the form \(\sum _{n \geqslant 1} \frac{f (\pi n \tau )}{n^s}\), 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.

Keywords

Trigonometric Dirichlet series Eichler integrals 

Mathematics Subject Classification

Primary 11F11 33E20 Secondary 11L03 33B30 

Notes

Acknowledgments

I thank Florian Luca for sharing his insight into his proof [11, Theorem 1] of the convergence of the secant Dirichlet series, and I am grateful to Bruce Berndt for helpful comments on an early version of this paper. I also thank the referee for suggestions improving the presentation.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignChampaignUSA

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