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\(p\)-adic polylogarithms and irrationality. (English) Zbl 1250.11070
According to a result by T. Rivoal [“Indépendance linéaire des valeurs des polylogarithmes”, J. Théor. Nombres Bordx. 15, No. 2, 551–559 (2003; Zbl 1079.11038)], for any rational number \(x\) such that \(|x|<1\), the \(\mathbb Q\)–space spanned by the set \(\{{\mathrm{Li}}_s(x)\}_{s\geq 1}\) of values of the complex polylogarithm \( {{\mathrm{Li}}}_s(x)=\sum_{k=1}^\infty x^k / k^s\) has infinite dimension. In the paper under review, the author considers the \(p\)–adic polylogarithm function, which he denotes by \({{\mathcal{L}\mathrm{i}}} _s(x)\), defined for an integer \(s\) and a \(p\)-adic number \(x\) with \(|x|_p<1\) by the same series. For an algebraic number \(\delta\) with \(|\delta|_p>1\) and an integer \(A\geq 2\), he gives a lower bound for the dimension of the \(\mathbb Q\)-space spanned by the values \(\{{{\mathcal{L}\mathrm{i}}} _s(1/\delta)\}_{1\leq s\leq A}\). As an example of his main result, he deduces the irrationality of the values \({{\mathcal{L}\mathrm{i}}} _2(234281)\in \mathbb Q_{234281}\) and \({{\mathcal{L}\mathrm{i}}} _2(2^{18})\in \mathbb Q_2\). The proof uses a criterion for linear independence which is a \(p\)-adic analog of a complex criterion due to Yu. V. Nesterenko and R. Marcovecchio [“Linear independence of linear forms in polylogarithms”, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 5, No. 1, 1–11 (2006; Zbl 1114.11063)]; see also A. Chantanasiri, [“Généralisation des critères pour l’indépendance linéaire de Nesterenko, Amoroso, Colmez, Fischler et Zudilin”, Ann. Math. Blaise Pascal 19, No. 1, 75–105 (2012; Zbl 1252.11056)] as well as explicit simultaneous Padé approximants of polylogarithms.
11J72 Irrationality; linear independence over a field
11J61 Approximation in non-Archimedean valuations
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