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A simple series representation for Apéry's constant

Published online by Cambridge University Press:  23 January 2015

John Melville
John Melville*
Affiliation:
56 North Meggetland, Edinburgh EH14 1XQ
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Extract

Apéry's constant is the value of ζ (3) where ζ is the Riemann zeta function. Thus

This constant arises in certain mathematical and physical contexts (in physics for example ζ (3) arises naturally in the computation of the electron's gyromagnetic ratio using quantum electrodynamics) and has attracted a great deal of interest, not least the fact that it was proved to be irrational by the French mathematician Roger é and named after him. See [1,2].

Numerous series representations have been obtained for ζ (3) many of which are rather complicated [3]. é used one such series in his irrationality proof. It is not known whether ζ (3) is transcendental, a question whose resolution might be helped by a study of an appropriate series representation of ζ (3).

Type
Articles
Information
The Mathematical Gazette , Volume 97 , Issue 540 , November 2013 , pp. 455 - 460
Copyright
Copyright © The Mathematical Association 2013

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References

1. Apéry, Roger, Irrationalite de ζ(2) and ζ(3), Astérisque 61 (1979), pp.11–13.Google Scholar
2. van der Poorten, A., A proof that Euler missed … Apéry's proof of the irrationality of ζ (3), The Mathematical Intelligencer 1 (4), (1979), pp.196–203.Google Scholar
3. Finch, S. R., Mathematical constants, Cambridge University Press (2003), pp. 40–43.Google Scholar