Remarks on linear independence of $q$-harmonic series

For any rational integer $q$, $|q|>1$, the linear independence over $\mathbb Q$ of the numbers $1$, $\zeta_q(1)$, and $\zeta_{-q}(1)$ is proved; here $\zeta_q(1)=\sum_{n=1}^\infty\frac1{q^n-1}$ is so-called $q$-harmonic series or $q$-zeta-value at the point $1$. Besides this, a measure of linear independence of these numbers is established.