Talk:Apéry's theorem

Merge?
The sole reason for Apéry's constant being named so is because he proved Apéry's theorem. It makes sense to merge this with Apéry's constant.


 * I have done the merger and removed the tag. Kcordina 16:28, 13 March 2006 (UTC)


 * I think this theorem is important enough to warrant its own page. Moreover it's one of the few theorems with any kind of story behind the proof, which I hope the current article captures Chenxlee (talk) 20:27, 10 May 2008 (UTC)

Zeta(2)
I don't understand about zeta(2) needing to be worked on as well - in what way? Clearly zeta(2) is irrational; it's been known to be pi^2/6 since Euler, and pi has been known to be transcendental since Lindemann. —Preceding unsigned comment added by 41.185.115.143 (talk) 21:52, 23 October 2010 (UTC)


 * Apéry and Beuker extended their proofs to cover ζ(2) as well for a couple of reasons. Firstly, maths isn't just about getting a proof for every result then moving onto the next one, it's also desirable to have the nicest possible proof of any given result.  Proving that ζ(2) is π²/6 and that π is irrational is harder and less elegant than Beuker's single integral proof of ζ(2)'s irrationality.  Secondly, it was hoped at the time that what works for ζ(3) should generalise to ζ(5), ζ(7), and so on.  Using the same tools to prove the irrationality of ζ(2) is easier but might have suggested a general method for proving the irrationality of the higher zeta constants too.  Admittedly, it didn't work out that way but it was still a strategy worth exploring. Chenxlee (talk) 10:47, 25 October 2010 (UTC)

Some suggested additions/ corrections
(1) Apéry published two papers on his zeta(3) result. The article has referenced the very cryptic 1979 Astérisque paper, but not the longer paper Interpolation de fractions continues et irrationalité de certaines constantes, Bulletin de la section des sciences du C.T.H.S III (1981), 37--53. This should be referenced as it explains his ideas in more detail.

(2) It is not correct to say that 'The starting point for Apéry was the series representation of ζ(3) as .........  '.  It is clear from the 1981 paper that the starting point was the continued fraction representation of the 'standard' series representation of zeta(3) as a sum of inverse cubes. Apéry discovered a method of obtaining from this a second, more rapidly converging continued fraction for zeta(3). The process can be iterated indefinitely. If the convergents of these continued fractions are denoted p(n,0)/q(n,0); p(n,1)/q(n,1) and so on, the diagonal elements p(n,n)/q(n,n) are also convergents of a rapidly converging continued fraction for zeta(3). The numerators and denominators of these latter convergents, with a common factor of (n!)^6 removed, are the quantities denoted v(n) and u(n) respectively in Apéry's Astérisque paper and as a(n) and b(n) in the Wikipedia article. This method of Apéry is explained in some detail in a paper by C.Batut and M.Olivier 'Séminaire de Théorie des Nombres de Bordeaux Année 1979-1980 - exposé No 23', a pdf of which can be readily accessed through the web via the Göttinger Digitalisierungszentrum.

The series to which the article refers (which was actually first obtained by Markov around 1890 by a different convergence acceleration technique) has as its partial sums the convergents p(n,n)/q(n,n), but this sum emerges as a by-product of Apéry's method, rather than being its origin.

(3) It is a little unjust to Apéry to say that van der Poorten and others filled 'gaps in his (Apéry's) proof'. Certainly van der Poorten in his Mathematical Intelligencer article does not claim to have done this and gives full credit to Apéry. Although Apéry was deliberately mysterious in the way he communicated his results, there seems little doubt that by the time of his June 1978 presentation he had validated the key steps. This wording of the article should be adjusted accordingly. —Preceding unsigned comment added by 58.164.29.81 (talk) 09:39, 31 March 2011 (UTC)


 * I've incorporated your points (1) and (3) but I don't currently have access to that book with Apéry's 1981 paper in. If you can change the article accordingly that'd be great, otherwise I'll try to do it when I've got a hold of and had a look at that paper.  Thanks. Chenxlee (talk) 14:11, 22 August 2011 (UTC)

Chenxlee: Thanks for your reply, which I just noticed. Do you have an email address to discuss changes? Or alternatively you can email me on cliffbott at hotmail dot com — Preceding unsigned comment added by 60.229.8.11 (talk) 02:43, 12 October 2011 (UTC)

Claim
Yong-Cheol Kim claims to have proved that zeta(5) is irrational. Doubt has been cast on this. — Preceding unsigned comment added by 93.97.194.200 (talk) 09:35, 20 August 2011 (UTC)
 * See http://arxiv.org/abs/1105.0730 — Preceding unsigned comment added by 81.139.120.242 (talk) 11:45, 22 August 2011 (UTC)
 * Kim's work has not been confirmed independently. — Preceding unsigned comment added by 81.148.81.141 (talk) 13:22, 23 August 2011 (UTC)

I have sent two messages to Dr. Kim, pointing out some mistakes in his proof. He answered me six months ago saying that he was trying to correct those steps, but without success. [F. M. S. Lima, University of Brasilia, 09/15/2012]. — Preceding unsigned comment added by 189.11.237.1 (talk) 16:33, 15 September 2012 (UTC)

Formula for c(n,k)
There is a problem with your formula for c(n,k) (which is taken directly from section 6 of van der Poorten's article): when k = 0 it is either undefined or infinite (depending on how one interprets the notation in the second sum). Consequently a(n) is either undefined or infinite for all n. — Preceding unsigned comment added by 121.216.78.221 (talk) 07:22, 12 October 2011 (UTC)
 * When k = 0 the second sum in the definition of cn,k is the empty sum so by convention has value zero; you're essentially adding up no things, which pretty much has to equal nothing. Chenxlee (talk) 13:01, 12 October 2011 (UTC)

Thanks - I was not aware of that convention. — Preceding unsigned comment added by 58.168.69.120 (talk) 23:22, 12 October 2011 (UTC)

Apery regarded as a crank
There was a huge level of skepticism that greeted his announcement of a proof that the number ζ(3) is irrational. The incredulity heightened when it was seen that the method of proof was very basic and used methods that could have been understood by mathematicians such as Euler who died nearly 200 years earlier. Many claimed proofs of old problems are rejected today at a glance simply because an experienced mathematician “knows” that methods that are too elementary can not solve the problem. All such avenues should have been explored before. So when he presented a lecture on his proof at the Journées Arithmétiques de Marseille it was greeted with doubt and disbelief. Each step he wrote on the blackboard appeared to be a remarkable identity that his audience considered unlikely to be true. When someone asked him “where do these identities come from?” he replied “They grow in my garden.” which obviously did not boost anyone’s confidence.

This information needs to be included in the article, as it is the truth.

[F. M. S. Lima, University of Brasilia]: Indeed, this and other features are confirmed in Chap.5 of the recently published book "The Irrationals", by Havil (see, e.g., http://www.amazon.com/Irrationals-Story-Numbers-Cant-Count/dp/0691143420 ).

82.21.204.60 (talk) 23:03, 24 April 2012 (UTC)