Mediterranean Journal of Mathematics

, Volume 13, Issue 2, pp 517–526 | Cite as

Evaluation of Series Involving the Product of the Tail of \({\zeta(k)}\) and \({\zeta(k+1)}\)

  • Ovidiu Furdui
  • Cornel Vălean


We aim at evaluating the following class of series involving the product of the tail of two consecutive zeta function values
where \({k\geq 2}\) is an integer. We show that the series can be expressed in terms of Riemann zeta function values and a special integral involving a polylogarithm function.

Mathematics Subject Classification

11M06 33B15 33E20 65B10 


Abel’s summation formula Polylogarithm function Riemann zeta function 


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  1. 1.
    Bonar, D.D., Koury, M.J.: Real Infinite Series, MAA, Washington DC, 2006Google Scholar
  2. 2.
    Choi J., Srivastava H.M.: Explicit Evaluations of Euler and Related Sums. Ramanujan J. 10, 51–70 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Freitas P.: Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums. Math. Comput. 74(251), 1425–1440 (2005)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Finch S.R.: Mathematical Constants, Encyclopedia of Mathematics and Its Applications 94. Cambridge University Press, Cambridge (2003)Google Scholar
  5. 5.
    Furdui O.: Limits, Series and Fractional Part Integrals. Problems in Mathematical Analysis. Springer, New York (2013)CrossRefMATHGoogle Scholar
  6. 6.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Elsevier Academic Press, Amsterdam (2007)Google Scholar
  7. 7.
    Srivastava H.M., Choi J.: Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht (2001)CrossRefMATHGoogle Scholar
  8. 8.
    Srivastava H.M., Choi J.: Zeta and q-Zeta Functions And Associated Series And Integrals. Elsevier, Amsterdam (2012)MATHGoogle Scholar
  9. 9.
    Whittaker E.T., Watson G.N.: A Course of Modern Analysis, Fourth Edition. Cambridge University Press, Cambridge (1927)Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsTechnical University of Cluj-NapocaCluj-NapocaRomania
  2. 2.Teremia MareRomania

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