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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)].

MSC:
 11J72 Irrationality; linear independence over a field 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
Citations:
Zbl 1015.11033; Zbl 1156.11327
Full Text: