The Ramanujan Journal

, Volume 46, Issue 1, pp 91–102 | Cite as

Infinite series representations for Dirichlet L-functions at rational arguments

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Abstract

With the help of transformation formulas of Dirichlet L-series, we generalize some classical formulas for the values \(\zeta (2N+1)\) given by Ramanujan. This will be done by constructing generalized Dirichlet series of the form \(\sum \nolimits _{n=1}^\infty a_n n^{-s/b}\) where \(b > 0\) is an integer, which have similar transformation properties as Dirichlet L-functions and by considering their Mellin transforms using contour integration methods.

Keywords

Dirichlet L-functions Eichler integral Mellin transform Infinite series 

Mathematics Subject Classification

11M99 

Notes

Acknowledgements

The author is grateful to Winfried Kohnen for many helpful comments, inspirations, and discussions, as well as the referee for useful suggestions which improved the paper.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Ruprecht-Karls UniversitätHeidelbergGermany

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