One of the eight numbers \(\zeta(5),\zeta(7),\cdots,\zeta(17),\zeta(19)\) is irrational
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)].
irrationality of values of the zeta-function
scientific article; zbMATH DE number 1927790