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Irrationality of at least one of the nine numbers \(\zeta(5),\zeta(7),\dots,\zeta(21)\). (Irrationalité d’au moins un des neuf nombres \(\zeta(5), \zeta(7),\dots,\zeta(21)\).) (French) Zbl 1015.11033
The author proves the irrationality of (at least) one of the nine numbers \(\zeta(2n+1)\) for \(n\in \mathbb{Z}\) and \(2\leq n\leq 10\). The method is analogous to the previous works of the author [C. R. Acad. Sci., Paris, Sér. I Math. 331, 267-270 (2000; Zbl 0973.11072); with K. Ball, Invent. Math. 146, 193-207 (2001; Zbl 1058.11051)]. The main differences are the choice of the series used and the use of the saddle point method for estimating a complex integral. Notice that W. Zudilin, with more refined calculations, obtained the irrationality of one of the four numbers \(\zeta(2n+1)\) for \(2\leq n\leq 5\). The nice survey by S. Fischler [Irrationalité de valeurs de zêta, Séminaire Bourbaki, Novembre 2002, Exposé No. 910)] is available at the address www.bourbaki.ens.fr/sem.precedents.html before its publication in Astérisque.

MSC:
11J72 Irrationality; linear independence over a field
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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