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New irrationality measures for $ q$-logarithms

Author(s): Tapani Matala-aho; Keijo Väänänen; Wadim Zudilin.
Journal: Math. Comp. 75 (2006), 879-889.
MSC (2000): Primary 11J82, 33D15
Posted: December 20, 2005
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Abstract: The three main methods used in diophantine analysis of $ q$-series are combined to obtain new upper bounds for irrationality measures of the values of the $ q$-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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Additional Information:

Tapani Matala-aho
Affiliation: Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulu, Finland
Email: tma@sun3.oulu.fi

Keijo Väänänen
Affiliation: Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulu, Finland
Email: kvaanane@sun3.oulu.fi

Wadim Zudilin
Affiliation: Department of Mechanics and Mathematics, Moscow Lomonosov State University, Vorobiovy Gory, GSP-2, 119992 Moscow, Russia
Email: wadim@ips.ras.ru

DOI: 10.1090/S0025-5718-05-01812-0
PII: S 0025-5718(05)01812-0
Received by editor(s): June 16, 2004
Received by editor(s) in revised form: March 10, 2005
Posted: December 20, 2005
Additional Notes: This work is supported by an Alexander von Humboldt research fellowship and partially supported by grant no.~03-01-00359 of the Russian Foundation for Basic Research
Copyright of article: Copyright 2005, American Mathematical Society

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