Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?
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:
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?
nt.number-theory soft-question riemann-zeta-function proof-theory irrational-numbers
YCor
KlangenKlangen
$\begingroup$ 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. $\endgroup$ – Vesselin Dimitrov Sep 26 '18 at 11:58
$\begingroup$ 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 $\zeta_3(3)$. $\endgroup$ – Vesselin Dimitrov Sep 26 '18 at 12:09
$\begingroup$ @VesselinDimitrov Very interesting, thank you for the references! $\endgroup$ – Klangen Sep 27 '18 at 9:16
$\begingroup$ @RichardEager : Does Zudilin really use ideas of Apéry? I had the impression that his methods (and Rivoal's) were not based on Apéry's methods. $\endgroup$ – Timothy Chow Sep 27 '18 at 14:35
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.
Timothy ChowTimothy Chow
As Frits Beukers write 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:
In the study of the moments
$$W_n(s)=\int\limits_{[0,1]^n}\left |\sum_{k=1}^ne^{2\pi i x_k}\right|d\vec{x}$$ 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 ) $$W_3(2k)=\sum_{j=0}^k\binom{k}{k}^2\binom{2j}{j},$$ and $$W_4(2k)=\sum_{j=0}^k\binom{k}{k}^2\binom{2j}{j}\binom{2(k-j)}{k-j}.$$ 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 Beuker's "Irrationality Proofs Using Modular Forms": http://carmasite.newcastle.edu.au/wadim/zw/Beukers-Asterisque1987.pdf
Positivity of rational functions and their diagonals (an article by Armin Straub and Wadim Zudilin): https://arxiv.org/abs/1312.3732
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
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).
Zurab SilagadzeZurab Silagadze
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