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

W. Zudilin

Zapiski Nauchnykh Seminarov POMI, Proceedings on number theory, Tome 322 (2005), pp. 107-124

Résumé

We construct simultaneous rational approximations to the $q$-series $L_1(x_1;q)$ and $L_1(x_2;q)$, and, if $x=x_1=x_2$, to the series $L_1(x;q)$ and $L_2(x;q)$, where \begin{gather*} L_1(x;q)=\sum_{n=1}^\infty\frac{(xq)^n}{1-q^n}=\sum_{n=1}^\infty\frac{xq^n}{1-xq^n}, \\ L_2(x;q)=\sum_{n=1}^\infty\frac{n(xq)^n}{1-q^n}=\sum_{n=1}^\infty\frac{xq^n}{(1-xq^n)^2}. \end{gather*} Applying the construction, we obtain quantitative linear independence over $\mathbb Q$ of the numbers in the following collections: $1$, $\zeta_q(1)=L_1(1;q)$, $\zeta_{q^2}(1)$, and $1$, $\zeta_q(1)$, $\zeta_q(2)=L_2(1;q)$ for $q=1/p$, $p\in\mathbb Z\setminus\{0,\pm1\}$.