One of the numbers \(\zeta(5)\), \(\zeta(7)\), \(\zeta(9)\), \(\zeta(11)\) is irrational. (English. Russian original) Zbl 1047.11072

Zudilin, V. V.

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.

Reviewer: Tanguy Rivoal (Caen)

MSC:

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

Keywords:

Riemann zeta-function; irrationality