On a permutation group related to \(\zeta (2)\). (English) Zbl 0864.11037

Rhin, Georges; Viola, Carlo

Acta Arith. 77, No. 1, 23-56 (1996).

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 \((h+i,i+j,j+k,k+\ell,\ell+h)\).

Reviewer: M.Waldschmidt (Paris)

MSC:

11J82

Measures of irrationality and of transcendence

Keywords:

permutation group; \(\zeta(2)\); \(p\)-adic valuations; JFM 35.0320.01; Riemann zeta-function; irrationality measure

Citations:

Zbl 0776.11036; Zbl 0830.11026; JFM 35.0320.01