The Ramanujan Journal

, Volume 48, Issue 3, pp 477–494 | Cite as

Series representations for the Apery constant \(\zeta (3)\) involving the values \(\zeta (2n)\)

  • Cezar LupuEmail author
  • Derek Orr


In this note, using the well-known series representation for the Clausen function, we also provide some new representations of Apery’s constant \(\zeta (3)\). In addition, by an idea from De Amo et al. (Proc Am Math Soc 139:1441–1444, 2011) we derive some new rational series representations involving even zeta values and central binomial coefficients. These formulas are expressed in terms of odd and even values of the Riemann zeta function and odd values of the Dirichlet beta function. In particular cases, we recover some well-known series representations of \(\pi \).


Riemann zeta function Clausen integral Rational zeta series representations Apery’s constant 

Mathematics Subject Classification

Primary 41A58 41A60 Secondary 40B99 



We would like to thank Piotr Hajlasz, Bogdan Ion, George Sparling and William C. Troy for some fruitful conversations which led to an improvement of the present paper.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA

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