On the functional transcendence of \(q\)-zeta values. (English. Russian original) Zbl 1093.11055

Math. Notes 73, No. 4, 588-589 (2003); translation from Mat. Zametki 73, No. 4, 629-630 (2003).
From the text: For any integer \(k\geq 1\), the power series \[ \zeta_q (k+1)= \sum^\infty_{n=1}\sigma_k(n)q^n,\quad\sigma_k(n)= \sum_{d|n}d^k,\tag{2} \] determines a \(q\)-extension of the value \(\zeta (k+1)\) of the Riemann zeta-function V. V. Zudilin [Math. Notes 72, No. 6, 858–862 (2002); translation from Mat. Zametki 72, No. 6, 936–940 (2002; Zbl 1044.11066)]. Moreover, the series in (1) is also meaningful for \(k=0\). By virtue of the trivial estimates \[ \sigma_k(n)\leq n^k\sum_{d|n}1\leq n^{k+1)}, \] this series represents an analytic function inside the unit disc for each integer \(k\geq 0\). The objective of this paper is to prove that the function \(\zeta_q (k+ 1)\) is not algebraic for any \(k\geq 1\). This (and even a stronger) result is well known for \(\zeta_q(2)\), \(\zeta_q(4)\), \(\zeta+q(6),\dots\), because the functions \(1+c_k\zeta_q(k)\) with suitable \(c_k\in\mathbb{Q}\) are Eisenstein series for each even \(k\geq 2\).
Theorem. For each \(k\geq 0\), the function \(\zeta_q(k+1)\) analytic on the domain \(|q|<1\) is transcendental over \(\mathbb{C}(q)\). In essence, this result is an application of problems from [G. Pólya and G. Szegő, Aufgaben und Lehrsätze aus der Analysis, 2. Band Funktionstheorie. Nullstellen. Polynome. Determinante. Zahlentheorie, Springer-Verlag, Heidelberg-Berlin (1925; JFM 51.0173.01), Division 8].


11J91 Transcendence theory of other special functions
Full Text: DOI