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**Irrationality proofs using modular forms.**
*(English)*
Zbl 0613.10031

Journées arithmétiques, Besançon/France 1985, Astérisque 147/148, 271-283 (1987).

[For the entire collection see Zbl 0605.00004.]

Partly from the author’s introduction: In the years following Apéry’s discovery of his irrationality proofs for \(\zeta\) (2), \(\zeta\) (3) it has become clear that these proofs do not only have significance as irrationality proofs, but the numbers that occur in them serve as interesting examples for several phenomena in algebraic geometry and modular form theory (see the citations in the paper). Furthermore it turns out that Apéry’s proofs themselves are simple consequences of elementary complex analysis on spaces of certain modular forms. Although the use of modular forms looks promising at first sight, the yield of new irrationality proofs is disappointingly low. However it is easy to overlook some simple tricks that may give new interesting results.

This paper describes the general framework of the proofs and gives, using the present approach, a very detailed proof of the irrationality of \(\zeta\) (3) as well as more summary proofs of additional results. This work goes some considerable distance towards giving a sophisticated answer to the reviewer’s question about Apéry’s proofs [Math. Intell. 1, 195-203 (1979; Zbl 0409.10028)]: ”What on earth is going on here?”

Partly from the author’s introduction: In the years following Apéry’s discovery of his irrationality proofs for \(\zeta\) (2), \(\zeta\) (3) it has become clear that these proofs do not only have significance as irrationality proofs, but the numbers that occur in them serve as interesting examples for several phenomena in algebraic geometry and modular form theory (see the citations in the paper). Furthermore it turns out that Apéry’s proofs themselves are simple consequences of elementary complex analysis on spaces of certain modular forms. Although the use of modular forms looks promising at first sight, the yield of new irrationality proofs is disappointingly low. However it is easy to overlook some simple tricks that may give new interesting results.

This paper describes the general framework of the proofs and gives, using the present approach, a very detailed proof of the irrationality of \(\zeta\) (3) as well as more summary proofs of additional results. This work goes some considerable distance towards giving a sophisticated answer to the reviewer’s question about Apéry’s proofs [Math. Intell. 1, 195-203 (1979; Zbl 0409.10028)]: ”What on earth is going on here?”

Reviewer: A.J.van der Poorten