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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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##### References:
 [1] Fischler, S.: Irrationality of values of $$L$$-functions of Dirichlet characters. Preprint arXiv:1904.02402 [math.NT] (2019) [2] Fischler, S.; Sprang, J.; Zudilin, W., Many odd zeta values are irrational, Compos. Math., 155, 938-952, (2019) · Zbl 1430.11097 [3] Krattenthaler, C.; Zudilin, W., Hypergeometry inspired by irrationality questions, Kyushu J. Math., 73, 189-203, (2019) [4] Rivoal, T.; Zudilin, W., Diophantine properties of numbers related to Catalan’s constant, Math. Ann., 326, 705-721, (2003) · Zbl 1028.11046 [5] Rivoal, T., Zudilin, W.: A note on odd zeta values. Preprint arXiv:1803.03160 [math.NT] (2018) [6] Sprang, J.: Infinitely many odd zeta values are irrational. By elementary means. Preprint arXiv:1802.09410 [math.NT] (2018) [7] Zudilin, W., Arithmetic of linear forms involving odd zeta values, J. Théor. Nombres Bordeaux, 16, 251-291, (2004) · Zbl 1156.11327 [8] Zudilin, W., One of the odd zeta values from $$\zeta (5)$$ to $$\zeta (25)$$ is irrational. By elementary means, SIGMA, 14, 028, (2018) · Zbl 1445.11063
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