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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.
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)
Full Text: DOI
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