On the approximation of the values of certain functions
Let \(\mathbb{I}\) be the field of rational numbers or an imaginary quadratic field, and \(\mathbb{Z}_{\mathbb{I}}\) be the ring of integers in the field \(\mathbb{I}\). Denote \[ f(z)= \sum^\infty_{\nu= 0}{z^\nu\over a^{2\nu}} \prod^\nu_{x= 1} {1\over x(x+\lambda)},\;f_1(z)= f(z),\;f_2(z)= azf'(z), \] where \(\lambda\in\mathbb{I}\), \(\lambda\neq -1,-2,\dots\); and \(a\in\mathbb{Z}_{\mathbb{I}}\), \(a\neq 0\) such that \(a\lambda\in\mathbb{Z}_{\mathbb{I}}\). Let \(\alpha= 1/c\), where \(c\in \mathbb{Z}_{\mathbb{I}}\), \(c\neq 0\). The author gives the best rational-fraction approximation to \(f_2(\alpha)/f_1(\alpha)\), i.e. proves that the inequality \[ |f_2(\alpha)/f_1(\alpha)- p/q|<|ac|^{-1}\log\log|q|/(|q|^2\log|q|) \] has infinitely many solutions \((p,q)\in \mathbb{Z}^2_{\mathbb{I}}\) with \(q\neq 0\); but for any \(\varepsilon> 0\) there is only a finite number of tuples \((p,q)\in\mathbb{Z}^2_{\mathbb{I}}\) with \(q\neq 0\) satisfying the inequality \[ |f_2(\alpha)/f_1(\alpha)- p/q|< (|ac|^{-1}- \varepsilon)\log\log|q|/(|q|^2\log|q|). \] Furthermore, the author also gives the simple explicit formulas of \(p\) and \(q\), which yield the best approximations.
scientific article; zbMATH DE number 1028436