Logarithms of algebraic numbers

Abstract

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.