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 \(\mathbb C(q)\).
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 \(\mathbb C(q)\).
Reviewer: Olaf Ninnemann (Berlin)
MSC:
| 11J85 | Algebraic independence; Gel’fond’s method |
| 11M41 | Other Dirichlet series and zeta functions |
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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