Linear and Algebraic Independence of $q$-Zeta Values

In the paper, results on linear and algebraic independence of $q$-series of the form $\zeta_q(s)=\sum_{n=1}^\infty\sigma_{s-1}(n)q^n$ over the field $\mathbb C(q)$ are obtained, where $\sigma_{s-1}(n)=\sum_{d\mid n}d^{s-1}$, $s=1,2,\dots$.