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Approximations to \(q\)-logarithms and \(q\)-dilogarithms, with applications to \(q\)-zeta values. (English) Zbl 1088.11052
J. Math. Sci., New York 137, No. 2, 4617-4633 (2006); translation from Zap. Nauchn. Semin. POMI 322, 107-124 (2005).
The author is able to construct explicit simultaneous approximations to the \(q\)-series \(L_1(x_i;q)\) for \(x_i\), \(i=1, 2, \dots,s\), and also to the \(q\)-series \(L_1(x;q)\) and \(L_2 (x;q)\), where \[ L_1(x;q)=\sum_{n=1}^\infty {(xq)^n \over 1-q^n}=\sum_{n=1}^\infty {xq^n \over 1-xq^n} \] and \[ L_2(x;q)=\sum_{n=1}^\infty {n(xq)^n \over 1-q^n}=\sum_{n=1}^\infty {xq^n \over (1-xq^n)^2}. \] These constructions have arithmetical corrolaries, in particular a quantitative linear indepence over \({\mathbb Q}\) for the three numbers \(1\), \(\zeta_q(1)=L_1(1;q)\) and \(\zeta_{q^2}(1)\), and also for the three numbers \(1\), \(\zeta_q(1)=L_1(1;q)\) and \(\zeta_{q}(2)=L_2(1;q)\) for \(q=1/p\) when \(p\in {\mathbb Z}\setminus \{0,\pm1\}\).

MSC:
11J82 Measures of irrationality and of transcendence
11M41 Other Dirichlet series and zeta functions
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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