- Apéry's theorem
In
mathematics , Apéry's theorem is a result innumber theory that states the number ζ(3) isirrational . That is, the number: cannot be written as a fraction "p"/"q".History
Euler proved in the eighteenth century that if "n" is a positive integer then we have:for some rational number "p"/"q". Specifically, writing the infinite series on the left as ζ(2"n") he showed:where the "Bn" are the rationalBernoulli numbers . Once it was proved that π"n" is always irrational this showed that ζ(2"n") is irrational for all positive integers "n".No such representation in terms of π is known for the so-called odd
zeta constants , the values ζ(2"n"+1) for positive integers "n", so it is not known if there are rational (or even algebraic) numbers ξ"n" such that:Because of this, no proof could be found to show that the odd zeta constants were irrational, even though they were - and still are - all believed to be transcendental. However, in June 1978
Roger Apéry gave a talk entitled "Sur l'irrationalité de ζ(3)." During the course of the talk he outlined proofs that ζ(3) and ζ(2) were irrational, the latter using methods simplified from those used to tackle the former rather than relying on the expression in terms of π. Due to the wholly unexpected nature of the result and Apéry's blasé and very sketchy approach to the subject many of the mathematicians in the audience dismissed the proof as flawed. Three of the audience members suspected Apéry was onto something, though, and set out to fill in the gaps in his proof.Two months later these three -
Henri Cohen ,Hendrik Lenstra , andAlfred van der Poorten - finished their work, and on August 18 Cohen delivered a lecture fleshing out the proof given by Apéry. Following the talk Apéry himself took to the podium to explain the source of some of his ideas. [A. van der Poorten, "A proof that Euler missed", Math. Intelligencer 1 (1979), pp. 195-203.]Apéry's Proof
Apéry's original proof [R. Apéry, "Irrationalité de ζ(2) et ζ(3)", Astérisque 61 (1979), pp. 11-13.] was based on the well known irrationality criterion from
Dirichlet , which states that a number ξ is irrational if there are infinitely many coprime integers "p" and "q" such that:for some fixed "c",δ>0.The starting point for Apéry was the series representation of ζ(3) as:Roughly speaking, Apéry then defined a sequence "cn,k" which converges to ζ(3) about as fast as the above series, specifically:He then defined two more sequences "an" and "bn" that, roughly, have the quotient "cn,k". These sequences were:and:The sequence "an"/"bn" converges to ζ(3) fast enough to apply the criterion, but unfortunately "an" is not an integer after "n"=2. Nevertheless, Apéry showed that even after multiplying "an" and "bn" by a suitable integer to cure this problem the convergence was still fast enough to guarantee irrationality.
Later Proofs
Within a year of Apéry's result an alternative proof was found by
Frits Beukers [F. Beukers, "A note on the irrationality of ζ(2) and ζ(3)", Bull. London Math. Soc. 11 (1979),pp. 268-272.] . His proof replaced Apéry's series with integrals involving theLegendre polynomials . Using a representation that would later be generalised toHadjicostas's formula , Beukers showed that:for some integers "An" and "Bn" and where "Pn"("x") is the "n"th Legendre polynomial. Using partial integration and the assumption that ζ(3) was rational and equal to "a"/"b", Beukers eventually derived the inequality:
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