The Ramanujan Journal

, Volume 15, Issue 3, pp 377–405 | Cite as

The evaluation of character Euler double sums

  • J. M. Borwein
  • I. J. ZuckerEmail author
  • J. Boersma


Euler considered sums of the form
Here natural generalizations of these sums namely
are investigated, where χ p and χ q are characters, and s and t are positive integers. The cases when p and q are either 1,2a,2b or −4 are examined in detail, and closed-form expressions are found for t=1 and general s in terms of the Riemann zeta function and the Catalan zeta function—the Dirichlet series L −4(s)=1s −3s +5s −7s +⋅⋅⋅ . Some results for arbitrary p and q are obtained as well.


Euler sums Dirichlet characters Riemann zeta function 

Mathematics Subject Classification (2000)

11M41 33E20 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Faculty of Computing ScienceDalhousie UniversityHalifaxCanada
  2. 2.Wheatstone Physics LaboratoryKing’s CollegeLondonUK
  3. 3.Dept. of Mathematics and Computing ScienceEindhoven University of TechnologyEindhovenThe Netherlands

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