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Apéry's proof of the irrationality of $\zeta(3)$ astounded contemporary mathematicians for its wealth of new ideas and techniques in proving the irrationality of a known constant. It is often the case that such new Theorems, whose substance contains completely novel approaches to an old problem, inspire solutions to other, similar problems by providing the tools necessary to tackle them.

However, in the case of Apéry's Theorem, it seems like the new techniques presented in the proof are not easily used within other contexts. This has led me to ask the question:

Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?

Since the proof contains numerous new ideas, I also put forward the following corollary question:

Which techniques employed in Apéry's proof of the irrationality of $\zeta(3)$ have been used within other proofs, whether in the field of irrationality/transcendence theory or other fields?

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    $\begingroup$ To a limited extent. I would recommend Beukers's paper Irrationality proofs using modular forms in Asterisque (1987), for two additional examples (Theorems 3 and 4) besides the old $\zeta(2) = \pi^2/6$, and for an illumination of what goes on behind Apéry's proof. $\endgroup$ Commented Sep 26, 2018 at 11:58
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    $\begingroup$ The idea of overconvergence, which Beukers accentuates in his first paragraph on the second page of his paper (this had been one of Apéry's apparent "miracles"), is a central topic in the subject of $p$-adic modular forms. Inspired by this paper of Beukers, Frank Calegari (Irrationality of certain $p$-adic periods for small $p$, IMRN 2005) made use of an overconvergence to prove the irrationality of the $2$-adic and $3$-adic zeta values $\zeta_2(3)$ and $\zeta_3(3)$. $\endgroup$ Commented Sep 26, 2018 at 12:09
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    $\begingroup$ @VesselinDimitrov Very interesting, thank you for the references! $\endgroup$
    – Klangen
    Commented Sep 27, 2018 at 9:16

4 Answers 4

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The proof of irrationality of $\displaystyle\sum_{n=0}^{+\infty}\frac1{F_n}$ (where $F_n$ is the $n$-th Fibonacci number) by RIchard André-Jeannin is an adaptation of the original Apery's proof of the irrationality of $\zeta(3)$.

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    $\begingroup$ For convenience, the reference is Richard André-Jeannin, “Irrationalité de la somme des inverses de certaines suites récurrentes”, Comptes Rendus de l'Académie des Sciences Série I, Math. 308 (1989), no. 19, 539–541. $\endgroup$ Commented Sep 3, 2023 at 16:19
  • $\begingroup$ Thank you for this reference, I was totally unaware of it. Also thanks to @hoboonsuan for the useful link. $\endgroup$
    – Klangen
    Commented Sep 4, 2023 at 7:03
  • $\begingroup$ Actually the André-Jeannin paper doesn't directly quote Apéry, but quotes: C. Batut, M. Olivier. Sur l'accélération de la convergence de certaines fractions continues, Séminaire de théorie des nombres de Bordeaux, 1979-1980, exposé 23. eudml.org/doc/182075 $\endgroup$
    – YCor
    Commented Sep 4, 2023 at 17:05
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Regarding your second question, Apéry's amazing formula $$\zeta(3) = {5\over 2} \sum_{n\ge1} {(-1)^{n-1} \over n^3 {2n \choose n}}$$ has inspired the search for analogous formulas for other zeta function values. I think that the earliest such was conjectured by Borwein and Bailey and proved by Almkvist and Granville: $$\sum_{k\ge0} \zeta(4k+3) z^{4k} = {5\over 2} \sum_{n=1}^\infty {(-1)^{n-1}\over n^3 {2n\choose n}} {1 \over 1-z^4\!/n^4} \prod_{m=1}^{n-1} {1 + 4z^4\!/m^4 \over 1 - z^4\!/m^4}.$$ There are other results along these lines; you can search for "Apéry-like" to locate the relevant literature. Most of these formulas were found empirically. Unfortunately, I don't think that any of these formulas have led to actual irrationality proofs. Apéry's original series converges fast enough to enable an irrationality proof, but the more complicated formulas that were found later have behavior that is not so easy to analyze.

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  • $\begingroup$ Thank you for this. Could you please provide a reference for the Almkvist & Granville proof? That would very much interest me. $\endgroup$
    – Klangen
    Commented Feb 3, 2022 at 19:48
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    $\begingroup$ The reference is Borwein and Bradley's Apéry-Like Formulae for ζ (4n+ 3), by Gert Almkvist and Andrew Granville, Experimental Mathematics 8 (1999), 197−203. $\endgroup$ Commented Feb 3, 2022 at 20:55
  • $\begingroup$ Thank you for that. $\endgroup$
    – Klangen
    Commented Feb 3, 2022 at 20:55
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As Frits Beukers writes in http://www.staff.science.uu.nl/~beuke106/caen.pdf "Ironically all generalisations tried so far did not give any new interesting results. Only through a combination of miracles such generalisations seem to work, which in practice means that we fall back to $\zeta(2)$ or $\zeta(3)$ again".

Nevertheless Apéry-like numbers do appear in somewhat unexpected places:

  1. In the study of the moments
    $$W_n(s)=\int\limits_{[0,1]^n}\left |\sum_{k=1}^ne^{2\pi i x_k}\right|^s d\vec{x}$$ of the distance traveled by a walk in the plane with unit steps in random directions. Namely for 3- and 4-step short random walks we have (see https://link.springer.com/article/10.1007/s11139-011-9325-y ) $$W_3(2k)=\sum_{j=0}^k\binom{k}{j}^2\binom{2j}{j},$$ and $$W_4(2k)=\sum_{j=0}^k\binom{k}{j}^2\binom{2j}{j}\binom{2(k-j)}{k-j}.$$ A striking feature that connects the 3- and 4-step random walk densities to irrationality proofs is their modularity: https://arxiv.org/abs/1103.2995 (Densities of short uniform random walks, by J.M. Borwein, A. Straub, J. Wan and W. Zudilin). See Beukers's "Irrationality Proofs Using Modular Forms": http://carmasite.newcastle.edu.au/wadim/zw/Beukers-Asterisque1987.pdf

  2. Positivity of rational functions and their diagonals (an article by Armin Straub and Wadim Zudilin): https://arxiv.org/abs/1312.3732

  3. In some series for $1/\pi$:https://icerm.brown.edu/materials/Slides/tw-14-5/Apery_numbers_and_their_experimental_siblings_]_Armin_Straub,_University_of_Illinois_at_Urbana-Champaign.pdf

  4. Calabi-Yau differential equations: https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/generalizations-of-clausens-formula-and-algebraic-transformations-of-calabiyau-differential-equations/D5A27594AC2F57F9B933A7716506C027 (Generalizations of Clausen's Formula and algebraic transformations of Calabi–Yau differential equations, by G. Almkvist, D. van Straten and W. Zudilin).

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    $\begingroup$ @GerryMyerson In the first display, the integrand is now raised to an $s$th power. In the other displays, $\binom{k}{k}$ are now corrected as $\binom{k}{j}$. The formula for $W_3(2k)$ is given in equation (18) of the paper "Densities of short uniform random walks" linked in the answer above. The numbers $W_4(2k)$ are identified, in the same paper, as Domb numbers (say, after Theorem 6), and their formula is as above. $\endgroup$ Commented Sep 4, 2023 at 16:19
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I also want to mention the formula (origin unknown) $$\dfrac{56\zeta(3)}{3}=\sum_{n\ge1}\dfrac{64^n}{n^3D_nD_{n-1}}\;,$$ where $D_n=W_4(2n)$ is the $n$th Domb number, equivalent to the continued fraction $$\zeta(3)=12/7/(2-16/(36-1024/(160-11664/(434-\dots))))$$ (numerators $-16n^6$ denominators $(2n-1)(5n^2-5n+2)$), which converges essentially like $4^{-n}$, as well as the formula $$\dfrac{7\zeta(3)}{2}=\sum_{n\ge1}\dfrac{16^n}{n^3C_nC_{n-1}}\;,$$ where $C_n=\sum_{n/2\le k\le n}\binom{n}{k}^2\binom{2k}{n}^2$ (I do not know if these have a name), equivalent to the continued fraction $$\zeta(3)=8/7/(1-1/(21-64/(95-729/(259-4096/(549-\dots)))))$$ (numerators $-n^6$ denominators $(2n-1)(3n^2-3n+1)$), which converges essentially like $(1+\sqrt{2})^{-4n}$.

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