## Diophantine problems for $$q$$-zeta values.(English. Russian original)Zbl 1044.11066

Math. Notes 72, No. 6, 858-862 (2002); translation from Mat. Zametki 72, No. 6, 936-940 (2002).
Consider the function of two variables $\zeta_q(k)=\sum_{n=1}^{\infty} n^{k-1}\frac{q^n}{1-q^{n}},$ for $$| q| <1$$, $$k=1, 2, \ldots$$, which is a $$q$$-analogue of the Riemann zeta function in the following sense: $\lim_{q\to 1} (1-q)^k \zeta_q(k)= (k-1)! \zeta(k).$ The author first presents some of his results about the diophantine results of $$\zeta_q(k)$$ when $$1/q\in\mathbb{Z}\setminus\{\pm 1\}$$: in particular, he outlines how he adapted the Rhin & Viola’s group method in the $$q$$-hypergeometric setting and how it enabled him to get remarkable new irrationality measures for $$\zeta_q(1)$$ and $$\zeta_q(2)$$.
He concludes with some remarks about the nature of $$\zeta_q(k)$$ as a function of $$q$$: when $$k\geq 4$$ is even, the functions $$\zeta_q(k)$$ are modular forms of weight $$k$$, hence they belong to $$\mathbb{Q}[\zeta_q(2), \zeta_q(4), \zeta_q(6)]$$ (which can be viewed as a functional $$q$$-analogue of $$\zeta(k)\in \mathbb{Q}\cdot \pi^k$$ for even $$k\geq 2$$), whereas nothing such is known for odd $$k\geq 1$$.
Since it is conjectured that $$\pi$$ and the numbers $$\zeta(2j+1)$$ ($$j\geq 1$$) are algebraically independent over $$\mathbb{Q}$$, the author proposes the following $$q$$-counterpart: prove that the functions $$\zeta_q(2), \zeta_q(4), \zeta_q(6)$$ and $$\zeta_q(2j+1)$$ ($$j\geq 0$$) are algebraically independent over the field $$\mathbb{C}(q)$$.

### MSC:

 11J82 Measures of irrationality and of transcendence 11J72 Irrationality; linear independence over a field

### Keywords:

irrationality measures
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