Apery for a Third Time

Unfolding Boy's Surface

We’d like to visit Roger Apery and his constant \zeta (3) once more.   Dragan Jankovic, our speaker in Math Club a few weeks ago, sent us an email pointing out that the controversy surrounding Apery’s announcement of his proof that that \zeta (3) is irrational, which we discussed here, has another side to the story.   It’s interesting to see the human side of mathematics, so we thought we should be fair to Apery and tell you the other side of things.

François Apery, Roger Apery’s son, wrote a brief biography which explains the context of the situation.  You can read the essay here.  The  short version is that Apery was a strong individualist in life and mathematics.  He annoyed more than a few people and that probably made them less friendly to Apery and his mathematical results (mathematicians are humans after all!).  Note: Bourbaki is a famous and very influential group of French mathematicians.  Normally it is a prestigious honor to be invited to be a member.  We’ll post more about this very interesting group sometime soon.

As François Apery writes:

[Apery’s] vision of mathematics was individualistic like his political philosophy, rebellious to all orthodoxy…. Practicing what he preached, he declined Dieudonné’s invitation to join Bourbaki.

The dominance of Bourbaki meant marginalization for the anti-Bourbakiste. Not being in sympathy even with all the other marginalized, Apéry eventually found himself nearly isolated.

[Apery] attacked the Lichnérowicz teaching reforms…. The reforms passed; he worried that 20 years later there would be a backslash in public opinion against mathematics, a prophecy that unfortunately came true. The instigators of the Lichnérowicz reform insisted on loyalty to their program and tried to brand any opposition to it as reactionary, which only hardened Apéry’s position and deepened his isolation in the community. It went so far that at the Journées Arithmétiques de Marseille in 1978, his lecture on the irrationality of \zeta (3) was greeted with doubt, disbelief, and then disorder.

— François Apéry (1996)

On the other hand, Roger Apery probably didn’t help things if the version of events described in this blog post by Dick Lipton is true:

I heard that when Apéry wrote on the board the key identity he needed,

\displaystyle  \zeta(3) = \frac{5}{2} \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{3} {2n \choose n}}

he gave a very strange answer to “where did this identity come from?” He is alleged to have answered, “they grow in my garden.” Obviously, this did not help make people feel comfortable. The identity is wonderful, the proof is correct

— Dick Lipton

P.S.  Francois Apery is a notable mathematician in his own right.  For example, he gave explicit equations using only polynomials for Boy’s Surface (many people were surprised that it could be done).   Here is an interesting article by Bruter about Boy’s Surface in art and architecture.

Boy's Surface

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5 thoughts on “Apery for a Third Time

  1. In this post of mine I computed in detail all algebraic steps of the derivation of the above series, as explained in section 3 of the Alf van der Poorten article [A. van der Poorten, A proof that Euler missed… Apéry’s proof of the irrationality of ζ(3) (An informal report), Math. Intelligencer 1:4 (1978/79), 195–203] This is not the central issue of your post but I here are the calculations (I think that the few Portuguese text will not inhibit the understanding of them). If you find that this is not the place to post these formulae, then please delete this post. In case someone needs a translation of the Portuguese text into English I can provide it (without giving any warranty of the full correctness of it, either grammatically, in the style or something else). “Estudei o artigo (*) de Alfred van der Poorten (actualmente Professor jubilado de Matemática) A proof that Euler Missed… Apery’s Proof of the Irrationality of , The Mathematical Intelligencer, Nº 1 (1979) pp. 195-203 (pdf), para o que necessitei de fazer alguns cálculos, dos quais apresento os do parágrafo 3. — Nota 1: uma soma telescópica de fracções racionais reais — Demonstração da identidade Fazendo e vem donde Por este motivo, mas, como comparando com (3), assim se completa a demonstração de (1). — Nota 2: um caso discreto particular da soma telescópica de fracções racionais — Dedução de Para e na identidade (1), vem, do lado esquerdo: e, do lado direito: como se queria deduzir, para . Para demonstrar falta, portanto, deduzir ou seja, simplificando Para o denominador do membro esquerdo vem, sucessivamente: Em resumo: e e, portanto donde, se obtem a identidade atrás, que se repete: — Nota 3: uma soma binomial — Dedução de [no original falta o factor 2 do segundo membro. No entanto, se fosse definido com este factor no denominador, a fórmula seria a que aparece no original.] , em que . Partindo desta definição temos e, sucessivamente mas como resulta donde somando ambos os membros, vem ou e, finalmente como se pretendia deduzir. — Nota 4: transformação noutra soma binomial equivalente — Dedução de Usando a notação de Iverson, o lado esquerdo pode escrever-se Esta notação significa neste caso Agora já podemos trocar a ordem dos dois somatórios O somatório do lado direito, no qual se utiliza a notação de Iverson, atendendo a que significa , escreve-se na notação habitual Provou-se que Mas como conclui-se que — Nota 5: somas parciais de — Dedução de Atendendo à definição de obtém-se Mas logo Agrupando agora os dois somatórios da mesma quantidade que figuram nos dois membros desta identidade, obtemos isto é — Nota 6: série equivalente a — Dedução de Vai-se mostrar que Em virtude de no intervalo se ter e , então ; donde resulta que e, portanto, , quando Se a soma dos termos em valores absolutos converge para então também o faz a própria soma. Fazendo agora em tender para obtém-se, como pretendíamos ” and a quote from Apéry’s words [Irrationalité de ζ(2) et ζ(3), Astérisque 61 (1979), 11–13]: “Pour étudier , nous posons: L’ utilisation de la diagonale donne la série qui à defaut de prouver immédiatement l’ irrationalité de converge mieux que . ” (*) A. van der Poorten, A proof that Euler missed… Apéry’s proof of the irrationality of ζ(3) (An informal report), Math. Intelligencer 1:4 (1978/79), 195–203
  2. It has been known for a long time that \zeta (s) is a rational of \pi^s, and hence irrational, if s is an even integer s\ge 2 (…). In contrast, the irrationality of \zeta (3) was proved by Roger Apéry only in 1979. Despite considerable effort the picture is rather incomplete about \zeta (s) for the other odd integers, s=2t+1\ge 5. Very recently, Keith Ball and Tanguy Rivoal proved that infinitely many of the values \zeta (2t+1) are irrational. (…) Wadim Zudilin has proved that at least one of the four values \zeta (5),\zeta (7),\zeta (9) and \zeta (11) is irrationational.

    Martin Aigner and Günter Ziegler Proofs from THE BOOK,

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