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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 $$L(s, \chi)$$
##### Keywords:
Riemann zeta-function; irrationality; saddle point method
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