On the linear independence of the values of polylogarithmic functions. (English) Zbl 0712.11040

In this very interesting paper the author deals with the arithmetic nature of values at rational points of the function \(L(\alpha,k;x)=\sum^{\infty}_{n=1}x^ n/(n-\alpha)^ k.\) A shortened version of the main theorem reads as follows. Let \(\alpha_ 1,...,\alpha_ m\) be rational numbers in [0,1) and \(k_ 1,...,k_ m\in {\mathbb{N}}\). Let \(\epsilon >0\). Then there exist (very explicit!) positive numbers \(q_ 0,\xi,H_ 0\) such that for any \(n_ 0,n_ 1,...,n_ k\in {\mathbb{Z}}\) with \(H=\max_{1\leq j\leq m}| n_ j| >H_ 0\) and any q, \(| q| >q_ 0\), we have \[ | n_ 0+\sum^{m}_{j=1}n_ jL(a_ j,k_ j;1/q)| \gg_{\epsilon}H^{-\xi -\epsilon}. \] Very nice applications are \(L_ 2(1/q)\not\in {\mathbb{Q}}\) for any \(q\in {\mathbb{Z}}\) with either \(q\geq 12\) or \(q\leq -8\), improvements of earlier results by Chudnovsky. The method consists of a very explicit construction of polynomials \(A_ n(x),B_{1,n}(x),...,B_{m,n}(x)\) of degree n satisfying \[ A_ n(1/x)L(\alpha_ j,k_ j;x)-B_{j,n}(1/x)=O(x^{[n,m]}). \] The polynomials \(A_ n\), which are explicitly written down, can be considered as generalisations of Legendre polynomials. At the end of the paper the author points out some very nice tricks to improve the values of \(q_ 0,\xi\).
Reviewer: F.Beukers


11J82 Measures of irrationality and of transcendence