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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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