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A criterion for the linear independence of polylogarithms over a number field. (English) Zbl 1430.11098
Summary: Let $$Li_s(z)$$ be the $$s$$–th polylogarithmic function. Let $$\alpha\in\overline{\mathbb Q}$$, $$0<\vert\alpha\vert<1$$. In this article we give a criterion for the linear independence of the $$s+1$$ numbers $$Li_1(z)$$, $$Li_2(z), \ldots, Li_s(z)$$ and $$1$$ over the number field $$\mathbb Q(\alpha)$$. the new part is that we prove the linear independence of such polylogarithms over a number field of arbitrary degree. We also show examples and a linear independence measure.

##### MSC:
 11J72 Irrationality; linear independence over a field