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Linear independence of dilogarithmic values. (English) Zbl 1454.11131

Summary: We establish the linear independence over \(\mathbb{Q}\), in both qualitative and quantitative forms, of the four numbers 1, \(\operatorname{Li}_{1}(1/z)=-\log(1-1/z)\), \(\operatorname{Li}_{2}(1/z)\) and \(\operatorname{Li}_{2}(1/(1-z))\), for all integers \(z\geq9\) or \(z\geq8\) and for rationals \(z=s/r\) or \(z=1-s/r\) with \(1<r<s\), where \(s\) is large in comparison with \(r\).

MSC:

11J72 Irrationality; linear independence over a field
11J82 Measures of irrationality and of transcendence
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