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Linear and algebraic independence of $$q$$-zeta values. (English. Russian original) Zbl 1160.11338
Math. Notes 78, No. 4, 563-568 (2005); translation from Mat. Zametki 78, No. 4, 608-613 (2005).
In this 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 $$\sigma_{s-1}(n) =\sum_{d\mid n} d^{s-1}$$, $$s= 1, 2,\dots$$. The author proves the following:
Theorem 2. For any $$s > 1$$, the functions $$\zeta_q(1)$$ and $$\zeta_q(s)$$ are algebraically independent over $$\mathbb C(q)$$.
Theorem 3. Let the functions $$\zeta_q(s_1),\zeta_q(s_2),\dots,\zeta_q(s_k)$$ be algebraically independent over $$\mathbb C(q)$$ for some number family $$s_1, s_2, \dots, s_k\in\mathbb N$$ such that $$s_i > 1$$, $$i = 1, \dots, k$$. In this case, the functions $$\zeta_q(1)$$ and $$\zeta_q(s_1),\zeta_q(s_2),\dots, \zeta_q(s_k)$$ are also algebraically independent over $$\mathbb C(q)$$.
Theorem 4. The functions $$\zeta_q(1)$$, $$\zeta_q(2)$$, $$\zeta_q(4)$$, and $$\zeta_q(6)$$ are algebraically independent over $$\mathbb C(q)$$.

##### MSC:
 11J85 Algebraic independence; Gel’fond’s method 11M41 Other Dirichlet series and zeta functions
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##### References:
 [1] K. Mahler, ”On algebraic differential equations satisfied by automorphic functions,” J. Austral. Math. Soc., 10 (1969), 445–450. · Zbl 0207.08302 · doi:10.1017/S1446788700007709 [2] V. V. Zudilin, ”Diophantine problems for q-zeta values,” Mat. Zametki [Math. Notes], 72 (2002), no. 6, 936–940. · Zbl 1044.11066 [3] Yu. V. Nesterenko, Transcendence of Some Functions [in Russian], Manuscript (2003).
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