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Apéry's theorem states that Apéry's constant $\zeta(3) := \sum_{n=0}^\infty n^{-3}$ is irrational. The Wikipedia article claims that this result was "wholly unexpected" to the point that the mathematical community initially did not believe it.

But for me personally, Apéry's theorem states exactly what I expected. Broadly speaking, I find that almost all "interesting" or "natural" series have an irrational sum (with pretty much the only exception being geometric series with a rational initial term and common coefficient). Indeed, as this question points out, it's arguably more natural to expect that Apéry's constant is a rational multiple of $\pi^3$ (and therefore irrational).

Why was the mathematical community so convinced that Apéry's constant was rational before Apéry published his proof to the contrary?

(Note: I don't really care exactly who believed what at the time. What I'm wondering is which mathematical heuristic(s) the community was using to conjecture that Apéry's constant was rational.)

Edit: the consensus view of the comments and answer is that the proof was unexpected, but the theorem itself was not. Since the standard usage of the word "result" refers to a theorem but not to a particular proof, I have edited the Wikipedia page to change the phrase "unexpected nature of the result" to "unexpected nature of the proof" in order to clarify the sentence's meaning.

tparker
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    nobody thought that it is rational; the unexpected thing was the proof – user8268 Jan 23 '21 at 23:15
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    @user8268 The Wikipedia article says "the unexpected nature of the result", not of the proof. – tparker Jan 23 '21 at 23:17
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    I imagine they meant the unexpected nature of it, insofar as who presented it and how. Apery was a good-but-not-great mathematician, claiming a result to a problem that had stumped Euler. He threw up a formula without proof early during the lecture he gave on it, joked instead of explaining its existence when poked, and people were immediately and rightfully very suspicious. – PrincessEev Jan 23 '21 at 23:25
  • People only paid attention when someone verified, to some reasonable error, the accuracy of that formula:

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

    A Numberphile video on this: link

     – PrincessEev Jan 23 '21 at 23:25
  • After all, it was Euler himself (if I recall) who showed the general formula for $\zeta(2n)$ and that, in particular, it involved a rational multiple of $\pi^{2n}$. So it would be very reasonable to expect $\zeta(3)$ to be somehow proportional to $\pi^3$ and thus irrational. I can't imagine why anyone would think it rational. – PrincessEev Jan 23 '21 at 23:26
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    " and Apéry's blasé and very sketchy approach to the subject, many of the mathematicians in the audience dismissed the proof as flawed" seems to be the important part of that sentence. – Will Jagy Jan 24 '21 at 01:41
  • Apery's proof is one of the inspiring examples (IMHO) that unsolved problems can be solved using simple ideas. I wish someone did something similar for FLT. – Paramanand Singh Jan 24 '21 at 08:14
  • This would be a better fit for hsm.se. – J.G. Jan 24 '21 at 21:45

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We can read in the Wikipedia article

  • ... However, in June 1978, Roger Apéry gave a talk titled Sur l'irrationalité de $\zeta(3)$. During the course of the talk he outlined proofs that $\zeta(3)$ and $\zeta(2)$ were irrational, the latter using methods simplified from those used to tackle the former rather than relying on the expression in terms of $\pi$. 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. ...

Here the phrase unexpected nature of the result addresses the unexpected technique to tackle the problem using methods known from showing the irrationality of $\zeta(2)$, while the experts rather expected methods similar to those which were used in trying to show that the quantities \begin{align*} \frac{\zeta(2n+1)}{\pi^{2n+1}} \end{align*} are transcendental.

At the end of this paragraph there is also a nice reference to the paper A proof that Euler missed... by A. v. d. Poorten. The author tells us:

  • ... I heard with some incredulity that, for one, Henri Cohen (Bordeaux, now Grenoble) believed that these claims might well be valid. Very much intrigued, I joined Hendrik Lenstra (Amsterdam) and Cohen in an evening's discussion in which Cohen explained and demonstrated most of the details of the proof. We came away convinced that Professeur Apéry had indeed found a quite miraculous and magnificient demonstration of the irrationality of $\zeta(3)$. ...

Conclusion: What we can read here addresses the interesting techniques to prove the irrationality of $\zeta(3)$. Irrationality was not seriously questioned.

Markus Scheuer
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