Hypergeometric rational approximations to \(\zeta (4)\). (English) Zbl 1460.11104

Marcovecchio, Raffaele; Zudilin, Wadim

Proc. Edinb. Math. Soc., II. Ser. 63, No. 2, 374-397 (2020).

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.

Reviewer: Jaroslav Hančl (Ostrava)

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)

Keywords:

irrationality measure; \(\pi^4\); hypergeometric function; hypergeometric integral; rational approximation; \(\zeta\)-function

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