Diophantine properties of numbers related to Catalan’s constant. (English) Zbl 1028.11046

Rivoal, T.; Zudilin, W.

Math. Ann. 326, No. 4, 705-721 (2003).

The main subject of this paper is to study the arithmetic nature of numbers related to Dirichlet’s \(\beta\)-function: \[ \beta(s)=\sum_{n=0}^\infty {(-1)^n\over (2n+1)^s}. \] Lindemann’s theorem implies that \(\beta(2n+1)\) is transcendental for \(n\in \mathbb N\). The authors consider the case of even integers and prove:
Theorem. Let \(a\) be an even positive integer and let \(\delta(a)\) be the dimension of the \(\mathbb Q\)-vector space spanned by the numbers 1, \(\beta(2)\), \(\beta(4)\), …, \(\beta(a)\), then \[ \delta(a) \geq {1+o(1)\over 2+\log(2)} \log (a) \qquad \text{ as} \;a \to \infty. \] Their method is effective and they show that at least one of the numbers \(\beta(2)\), \(\beta(4)\), …, \(\beta(14)\) is irrational. The main tool in the proof is the study of well-chosen hypergeometric functions. This study is quite delicate and detailed. The proof is about 15 pages long.

Reviewer: Maurice Mignotte (Strasbourg)

MSC:

11J72

Irrationality; linear independence over a field

Keywords:

Catalan’s constant; irrationality