Arithmetic of Catalan’s constant and its relatives. (English) Zbl 1451.11078

Zudilin, Wadim

Abh. Math. Semin. Univ. Hamb. 89, No. 1, 45-53 (2019).

Let \(\beta(s)=\sum_{k=0}^\infty \frac {(-1)^k}{(2k+1)^s}\) be the Dirichlet beta function. The author proves that at least one of the numbers \(\beta(2)\)-Catalan constant, \(\beta(4)\), \(\beta(6)\), \(\beta(8)\), \(\beta(10)\), \(\beta(12)\) is irrational. The proof is in the spirit of Hermite, namely, it evaluates the integral which is from one point of view a small number below one in absolute value and from the second point of view a non-zero integer.

Reviewer: Jaroslav Hančl (Ostrava)

MSC:

11J72	Irrationality; linear independence over a field
11Y60	Evaluation of number-theoretic constants
33C20	Generalized hypergeometric series, \({}_pF_q\)

Keywords:

irrationality; Catalan constant; Dirichlet beta function; hypergeometric series

References:

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