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On a permutation group related to $$\zeta (2)$$. (English) Zbl 0864.11037
For nonnegative integers $$h,i,j,k,\ell$$, define $I(h,i,j,k,\ell) =\int^1_0 \int^1_0 {x^h (1-x)^i y^k(1-y)^j \over (1-xy)^{i+j- \ell}} { dx dy \over 1-xy}.$ In 1905, A. C. Dixon showed that the value of this integral is unchanged under a cyclic permutation of $$h,i,j,k,\ell$$ [Proc. Lond. Math. Soc., II. Ser. 2, 8-15 (1905; JFM 35.0320.01)]. This property can also be shown by means of the birational mapping of the plane $$(x,y) \mapsto \left( {1-x \over 1-xy}, 1-xy\right)$$ which maps the unit square $$[0,1]^2$$ onto itself and has period 5; this transformation was introduced by the authors in [Ann. Inst. Fourier 43, 85-109 (1993; Zbl 0776.11036)].
In the paper under review, the authors show that this number $$I(h,i,j,k, \ell)$$ is of the form $$a-b \zeta(2)$$ for some $$a\in\mathbb{Q}$$ and $$b\in\mathbb{Z}$$. Further, they give a sharp upper bound for a denominator of the rational number $$a$$. Using the special values $$h=i= 12n$$, $$j=k=14n$$, $$\ell=13n$$, $$(n\geq 1)$$, they derive the irrationality measure 5.441243 for the transcendental number $$\zeta(2) =\pi^2/6$$. The previously best known result, due to M. Hata [J. Aust. Math. Soc., Ser. A 58, 143-153 (1995; Zbl 0830.11026)], was 5.687, and involved the sequence $$I(hn,in,jn,kn,\ell n)$$, $$(n\geq 1)$$, with parameters $$h=i=15$$, $$j=k=14$$, $$\ell=17$$.
The proof depends on sharp estimates for $$p$$-adic valuations, and this is achieved by means of a detailed study of the action of the permutations $\tau= (h+i,i+j,j+k,k+\ell,\ell+h),\;\;\sigma= (h+i,j+k) (k+\ell,\ell+h),\;\;\varphi= (i+j,\ell+h)$ on the 5 numbers $$(h+i,i+j,j+k,k+\ell,\ell+h)$$.

##### MSC:
 11J82 Measures of irrationality and of transcendence
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