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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
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