## 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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### References:

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