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One of the numbers $$\zeta(5)$$, $$\zeta(7)$$, $$\zeta(9)$$, $$\zeta(11)$$ is irrational. (English. Russian original) Zbl 1047.11072
Russ. Math. Surv. 56, No. 4, 774-776 (2001); translation from Usp. Mat. Nauk 56, No. 4, 149-150 (2001).
The author sketches a proof of the important result stated in the title: his method is based on a subtle refinement of the hypergeometric technique used by the reviewer to prove the weaker result that one of the nine numbers $$\zeta(5)$$, $$\zeta(7)$$, …, $$\zeta(21)$$ is irrational. A related preprint “Arithmetic of linear forms in odd zeta values” of the author contains all the necessary explanations and will appear in J. Théor. Nombres Bordx.

##### MSC:
 11J72 Irrationality; linear independence over a field 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
##### Keywords:
Riemann zeta-function; irrationality
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