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Arithmetic of Catalan’s constant and its relatives. (English) Zbl 1451.11078
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.
MSC:
11J72 Irrationality; linear independence over a field
11Y60 Evaluation of number-theoretic constants
33C20 Generalized hypergeometric series, \({}_pF_q\)
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