There are infinitely many irrational values of the Riemann zeta function at odd integers. (La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs.) (French) Zbl 0973.11072

Rivoal, Tanguy

C. R. Acad. Sci., Paris, Sér. I, Math. 331, No. 4, 267-270 (2000).

The author proves the following theorem: for every \(\varepsilon> 0\), there exists \(N(\varepsilon)> 0\) such that, for every \(n> N(\varepsilon)\), \[ \dim_{\mathbb{Q}} (\mathbb{Q}+ \mathbb{Q} \zeta(3)+ \cdots+ \mathbb{Q} \zeta(2n+1))\geq \frac{1-\varepsilon}{1+\log 2}\cdot \log n. \] As a consequence, the zeta-function takes infinitely many irrational values at odd integers. The proof uses Padé-approximants of type I for the polylogarithm functions [E. M. Nikishin, Mat. Sb. 37, 381-388 (1980; Zbl 0441.10031)] modified following an idea of K. Ball. The conclusion follows from a criterion of linear independence due to Yu. V. Nesterenko [Vestn. Mosk. Univ., Ser. I 1985, No. 1, 46-49 (1985; Zbl 0572.10027)].

Reviewer: D.Duverney (Lille)

MSC:

11J72	Irrationality; linear independence over a field
33B30	Higher logarithm functions
11M06	\(\zeta (s)\) and \(L(s, \chi)\)
41A21	Padé approximation

Keywords:

irrationality; Riemann zeta function; polylogarithm functions; Padé approximants

Citations:

Zbl 0414.10032; Zbl 0572.10027; Zbl 0441.10031