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Can polylogarithms at algebraic points be linearly independent? (English) Zbl 1459.11144
For \(0\le x\leq 1\) the \(s\)-th Lerch function is defined as \[\Phi_s(x,z)=\sum_{k=0}^{\infty}\frac{z^{k+1}}{(k+x+1)^s}\] for \(s=1,2,\ldots,r\). For \(x=0\) this is the polylogarithmic function Li\(_s(z)\).
For given pairwise distinct algebraic numbers \(\alpha_j\) with \(0\leq |\alpha_j|\leq 1\) \((1\le j\le m)\) the authors state a linear independence criterion over algebraic number fields of the numbers \(\Phi_i(x,\alpha_j)\), for \(1\le s\le r\), \(1\le j\le m\) and 1. An explicit sufficient condition is given for the linear independence of values of the Lerch functions \(\Phi_i(x,z)\), \(1\le s\le r\), at \(m\) distinct points in an algebraic number field of arbitrary finite degree without any assumptions on \(r\) and \(m\). For \(x=0\), these results imply the linear independence of polylogarithms of distinct algebraic numbers of arbitrary degree, subject to a metric condition.
The proof bases on Padé approximation techniques.
The paper is illustrated by interesting examples. For \(|b|\ge e^{2715}\) the numbers 1, Li\(_1(1/b)\),\ldots,Li\(_{10}(1/b)\),\ldots,Li\(_1(1/(10b))\),\ldots,Li\(_{10}(1/(10b))\) are linearly independent over \(\mathbb Q\).

11G55 Polylogarithms and relations with \(K\)-theory
11J72 Irrationality; linear independence over a field
11J82 Measures of irrationality and of transcendence
11J86 Linear forms in logarithms; Baker’s method
11M35 Hurwitz and Lerch zeta functions
11D75 Diophantine inequalities
11D88 \(p\)-adic and power series fields
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