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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.
MSC:
 11J72 Irrationality; linear independence over a field 11J61 Approximation in non-Archimedean valuations
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