One of the eight numbers \(\zeta(5),\zeta(7),\cdots,\zeta(17),\zeta(19)\) is irrational. (English. Russian original) Zbl 1022.11035

Math. Notes 70, No. 3, 426-431 (2001); translation from Mat. Zametki 70, No. 3, 472-476 (2001).
The author proves the result given in the title by refining the method used by the reviewer to prove the following weaker result: one of the nine numbers \(\zeta(5), \zeta(7), \dots, \zeta(21)\) is irrational [Acta Arith. 103, 157-167 (2002; Zbl 1015.11033)]. Using a very-well-poised series (of hypergeometric type), he constructs a sequence of linear forms \(S_n=p_{0,n}+p_{1,n}\zeta(5)+\cdots+p_{8,n}\zeta(19)\) with rational coefficients: his improvement is due to his very careful study of the \(p\)-adic valuation of the integers \(D_np_{j,n}\), where \(D_n\) denotes a common denominator \(D_n\) of the \(p_{j,n}\). He uses for this a “prime extracting” method introduced by Chudnovsky, Hata and others, and removes a “big” common factor \(\Pi_n\) of the integers \(D_np_{j,n}\). Finally, he applies the saddle point method to prove that \(\Pi_n^{-1}D_nS_n\) is never 0 and tends to 0 as \(n\) tends to infinity, which proves the theorem.
He also notes that the same method proves the irrationality of at least one number in each of the sets \(\{\zeta(7), \zeta(9),\dots, \zeta(35)\}\) and \(\{\zeta(9), \zeta(11),\dots, \zeta(51)\}\).
Note finally that, in the meantime, a further refinement of this arithmetical scheme has enabled the author to prove that at least one of the four numbers \(\zeta(5), \zeta(7), \zeta(9), \zeta(11)\) is irrational [“Arithmetic of linear forms involving odd zeta values”, (to appear in) J. Théor. Nombres Bordx. 16, No. 1, 251–291 (2004; Zbl 1156.11327)].


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