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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
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