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

MSC:
 11J72 Irrationality; linear independence over a field
Keywords:
Catalan’s constant; irrationality
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