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New irrationality measures for -logarithms

Authors:	Tapani Matala-aho Keijo Vä ä nä nen Wadim Zudilin

Institution:	Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulu, Finland ; Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulu, Finland ; Department of Mechanics and Mathematics, Moscow Lomonosov State University, Vorobiovy Gory, GSP-2, 119992 Moscow, Russia

Abstract:	The three main methods used in diophantine analysis of -series are combined to obtain new upper bounds for irrationality measures of the values of the -logarithm function $\displaystyle \ln _{q}(1-z)=\sum _{\nu =1}^{\infty }\frac{z^{\nu }q^{\nu }}{1-q^{\nu }}, \qquad \vert z\vert\leqslant 1,$ when $p=1/q\in \mathbb{Z}\setminus \{0,\pm 1\}$ and $z\in \mathbb{Q}$ .

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