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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
Article

Abstract

We aim at evaluating the following class of series involving the product of the tail of two consecutive zeta function values
$$\sum\limits_{n=1}^{\infty}\left(\zeta(k)-1-\frac{1}{2^k}-\cdots-\frac{1}{n^k}\right)\left(\zeta(k+1)-1-\frac{1}{2^{k+1}}-\cdots-\frac{1}{n^{k+1}}\right),$$
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 

Keywords

Abel’s summation formula Polylogarithm function Riemann zeta function 

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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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