On the irrationality measure of \(\pi^2\). (English. Russian original) Zbl 1286.11111

Zudilin, W.

Russ. Math. Surv. 68, No. 6, 1133-1135 (2013); translation from Usp. Mat. Nauk 68, No. 6, 171-172 (2013).

Summary: In this note we establish a new bound on the quality of rational approximations of \(\zeta(2)= \pi^2/6\):
Theorem 1. \(\mu(\zeta(2)\leq 5.09541178\ldots\).
Theorem 2. \(\mu(\pi\sqrt d)\leq 10.19082357\ldots\) for any \(d\in\mathbb Q\backslash 0\)..

MSC:

11J82	Measures of irrationality and of transcendence
11J72	Irrationality; linear independence over a field
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; irrationality exponent; hypergeometric approximation