On the irrationality of the values of the zeta function at odd integer points. (English. Russian original) Zbl 1037.11048

Zudilin, V. V.

Russ. Math. Surv. 56, No. 2, 423-424 (2001); translation from Usp. Mat. Nauk 56, No. 2, 215-216 (2001).

Generalizing the construction introduced by T. Rivoal [C. R. Acad. Sci., Paris, Sér. I, Math. 331, 267–270 (2000; Zbl 0973.11072)], the author establishes the following results:
Theorem 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.
Theorem 2: There is an odd integer \(a\leq 145\) such that 1, \(\zeta(3)\) and \(\zeta(a)\) are linearly independent over \(\mathbb{Q}\).
Theorem 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 \[ \delta(a)> 0.395 \log a> 2\log a/3 (1+\log 2). \] Theorem 2 improves 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.

Reviewer: Zhu Yaochen (Beijing)

MSC:

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

Keywords:

irrationality; linear independence; Riemann zeta-function

Citations:

Zbl 0973.11072