A Proof that Euler Missed: Evaluating ξ(2) the Easy Way

  • Tom M. Apostol


R. Apéry [1] was the first to prove the irrationality of
$$\zeta \left( 3 \right) = \sum\limits_{n = 1}^\infty {\frac{1}{{{n^3}}}} $$


Mathematical INTElLIGENCER Straightforward Manner Short Proof Multiple Integral Double Integral 
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  1. 1.
    R. Apéry (1979) Irrationalité de;(2) et;(3). Astérisque 61:11–13. Paris: Societé Mathématique de France.Google Scholar
  2. 2.
    F. Beukers (1979) A note on the irrationality of C(2) and C(3), Bull. Lon. Math. Soc. 11:268–272.Google Scholar
  3. 3.
    F. Goldscheider (1913) Arch. Math. Phys. 20: 323–324.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Tom M. Apostol
    • 1
  1. 1.California Institute of TechnologyPasadenaUSA

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