Logarithms of algebraic numbers

Kühne, Lars

Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 2, pp. 499-535.

Cet article est consacré à une nouvelle démonstration de la transcendance des évaluations du logarithme archimédien en tout nombre algébrique, exception faite de l’unité. Par analogie avec d’autres démonstrations de ce résultat, nous employons une variante de l’approximation de Padé pour le logarithme népérien. Une différence cependant : la construction de ces approximations de Padé est ici obtenue par le lemme de Siegel. La méthode exposée suggère des généralisations qui sont aussi évoquées.

This article is devoted to a new proof of transcendence for evaluations of the archimedean logarithm at all algebraic numbers except unity. As in other proofs of the same theorem, a sort of Padé approximation for the natural logarithm is employed. Whereas in previous approaches the used Padé approximants have been obtained rather ad hoc, we construct them here systematically by Siegel’s Lemma. The method presented suggests some generalizations, which are also briefly surveyed.

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