Lower bounds for linear forms in values of polylogarithms (Q2387880)

The main result of this paper concerns the polylogarithmic functions \(L_k(z), k\in \mathbb{N}\), defined in the unit circle by the power series \(\sum_{n=1}^\infty z^n/n^k\), and reads as follows. For \(a,b,m\in \mathbb{Z}\) with \(a,m>0\) and \(|b|>a^{m+1}e^{m(m+\log m)}\) there exists a constant \(c=c(a,b,m)>0\) such that, for all \((x_0,x_1,\dots ,x_m)\in \mathbb{Z}^{m+1}\setminus \{\underline{0}\}\) satisfying \(\overline{x}_1\geq\dots \geq \overline{x}_m\) (where \(\overline{x}_k\geq \max(1,|x_k|)\) for \(k=1,\dots ,m\)), the Baker type inequality \(|x_0+\sum_{k=1}^m x_kL_k(a/b)|> c\cdot (\overline{x}_1\cdots \overline{x}_m)^{-1}\cdot \overline{x}_1^{-\delta}\) holds with some \(\delta>0\), which is precisely given in terms of \(a,b,m\). The proof uses explicit constructions of Hermite-Padé approximations of the second kind. The theorem quoted above slightly refines an earlier result of the first author [Math. Notes 67, No. 3, 372--381 (2000); translation from Mat. Zametki 67, No. 3, 441--452 (2000; Zbl 1125.11334)], where she applied Padé approximations of the first kind.