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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.
##### MSC:
 11J72 Irrationality; linear independence over a field 11G55 Polylogarithms and relations with $$K$$-theory 41A21 Padé approximation
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##### References:
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