One of the eight numbers \(\zeta(5),\zeta(7),\cdots,\zeta(17),\zeta(19)\) is irrational
type: Item
instance of: scholarly article
title: One of the eight numbers \(\zeta(5),\zeta(7),\cdots,\zeta(17),\zeta(19)\) is irrational
zbMATH Open document ID: 1022.11035
DOI: 10.1023/A:1012312315852
published in: Mathematical Notes
publication date: 2003-06-15T00:00:00Z
review text: 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)].
reviewed by: T. Rivoal
Mathematics Subject Classification ID: 11J72
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 1927790
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: irrationality of values of the zeta-function
zbMATH Keywords: Nikishin-Rivoal arithmetical scheme
author: Wadim Zudilin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1927790