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Two hypergeometric tales and a new irrationality measure of \(\zeta (2)\). (English) Zbl 1307.11085

The author proves that the irrationality exponent of the number \(\zeta(2)=\pi^2/6\) is bounded from above by \(5.09541178\). He also proves several identities for hypergeometric integrals which include the number \(\zeta(2)\). The proof is not simple and requires some knowledge from the theory of the hypergeometrical functions.

MSC:

11J82 Measures of irrationality and of transcendence
11Y60 Evaluation of number-theoretic constants
33C20 Generalized hypergeometric series, \({}_pF_q\)
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
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