On the irrationality of the values of the zeta function at odd integer points (Q2784567)

Generalizing the construction introduced by \textit{T. Rivoal} [C. R. Acad. Sci., Paris, Sér. I, Math. 331, 267--270 (2000; Zbl 0973.11072)], the author establishes the following results:NEWLINENEWLINETheorem 1. Each of the sets \(\{\zeta(k)\), \(k=5,7, \dots, 21\}\), \(\{\zeta(k)\), \(k=7,9, \dots, 37\}\), \(\{\zeta(k)\), \(k= 9, 11, \dots, 53\}\) contains at least one irrational number.NEWLINENEWLINETheorem 2: There is an odd integer \(a\leq 145\) such that 1, \(\zeta(3)\) and \(\zeta(a)\) are linearly independent over \(\mathbb{Q}\).NEWLINENEWLINETheorem 3: Let \(a\geq 3\) be any odd integer, and let \(\delta(a)\) be the dimension of the space spanned over \(\mathbb{Q}\) by the numbers \(1, \zeta(3), \zeta(5), \dots, \zeta(a-2), \zeta(a)\), then NEWLINE\[NEWLINE\delta(a)> 0.395 \log a> 2\log a/3 (1+\log 2).NEWLINE\]NEWLINE Theorem 2 improves \textit{T. Rivoal}'s result [Rapport de recherche SDAD, No. 2000-9, Univ. Caen (2000)], in which \(a\leq 169\) is assumed. Proofs are only sketched.