Geodesic
Linear and Algebraic Independence of $q$-Zeta Values
Matematičeskie zametki, Tome 78 (2005) no. 4, pp. 608-613

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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 $\sigma_{s-1}(n)=\sum_{d\mid n}d^{s-1}$, $s=1,2,\dots$. 
@article{MZM_2005_78_4_a10,
     author = {Yu. A. Pupyrev},
     title = {Linear and {Algebraic} {Independence} of $q${-Zeta} {Values}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {608--613},
     publisher = {mathdoc},
     volume = {78},
     number = {4},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_78_4_a10/}
}
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Yu. A. Pupyrev. Linear and Algebraic Independence of $q$-Zeta Values. Matematičeskie zametki, Tome 78 (2005) no. 4, pp. 608-613. http://geodesic.mathdoc.fr/item/MZM_2005_78_4_a10/