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Journal of Mathematical Sciences

, Volume 137, Issue 2, pp 4673–4683 | Cite as

Approximations to q-logarithms and q-dilogarithms, with applications to q-zeta values

  • W. Zudilin
Article

Abstract

We construct simultaneous rational approximations to q-series L1(x1; q) and L1(x2; q) and, if x = x1 = x2, to series L1(x; q) and L2(x; q), where
$$\begin{gathered} L_1 (x;q) = \sum\limits_{n = 1}^\infty {\frac{{(xq)^n }}{{1 - q^n }}} = \sum\limits_{n = 1}^\infty {\frac{{xq^n }}{{1 - xq^n }}} , \hfill \\ L_2 (x;q) = \sum\limits_{n = 1}^\infty {\frac{{n(xq)^n }}{{1 - q^n }}} = \sum\limits_{n = 1}^\infty {\frac{{xq^n }}{{(1 - xq^n )^2 }}} . \hfill \\ \end{gathered} $$
. Applying the construction, we obtain quantitative linear independence over ℚ of the numbers in the following collections: 1, ζq(1) = L1(1; q), \(\zeta _{q^2 } \) and 1, ζq(1), ζq(2) = L2(1; q) for q = 1/p, p ε ℤ \ {0,±1}. Bibliography: 14 titles.

Keywords

Rational Number Linear Independence Simultaneous Approximation Basic Hypergeometric Series Lambert Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • W. Zudilin
    • 1
  1. 1.Department of Mechanics and MathematicsMoscow Lomonosov State UniversityMoscowRussia

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