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A note on linear independence of polylogarithms over the rationals. (English) Zbl 1286.11108
The paper contains a new lower bound for the dimension of the linear space over the rationals spanned by \(1\) and values of polylogarithms of a given rational number. As a special case the authors prove the following theorem.
Let \(\alpha=\frac{p}{q}\) be a rational number with \(0<|\alpha|<1\) and \(q\leq50\). Let \(s\geq356\). Then \[ \dim_{\mathbb Q}\bigl(\mathbb Q+\mathbb Q\, Li_1(\alpha)+\dots+\mathbb Q\, Li_s(\alpha)\bigr)\geq3 \] where \(Li\) is the polylogarithmic function.
11J72 Irrationality; linear independence over a field
11G55 Polylogarithms and relations with \(K\)-theory
41A21 Padé approximation
Full Text: DOI Euclid
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