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Hypergeometric rational approximations to \(\zeta (4)\). (English) Zbl 1460.11104
The main result of this paper is that \(\mu(\zeta(4))\leq 12.51085940\dots\) where \(\zeta(x)=\sum_{n=1}^\infty \frac 1{n^x}\) is a zeta function and \(\mu(z)\) is the irrationality measure exponent of the number \(z\). The proof makes use the hypergeometric integrals and group theory.
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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