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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
##### Keywords:
irrationality measure
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##### References:
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