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On the irrationality measure of \(\zeta (2)\). (English) Zbl 0776.11036
The authors provide an effective irrationality measure \(7.398\ldots\) for \(\zeta(2)\). This implies the irrationality measure \(14.796\ldots\) for \(\pi\). To obtain this measure the authors construct a sequence of approximations via integrals of the form \(\int_ 0^ 1\int_ 0^ 1 H(x,y)(1-xy)^{-n-1} dx dy\). The main part of the paper is devoted to the study of the asymptotics of this integral and to an optimal choice for \(H(x,y)\). In [Acta Arith. 63, 335-349 (1993; Zbl 0776.11033)]M. Hata gives the irrationality measure \(8.0161\) using an entirely different method.

MSC:
11J82 Measures of irrationality and of transcendence
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