zbMATH — the first resource for mathematics

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].
MSC:
 11J91 Transcendence theory of other special functions
Citations:
Zbl 1044.11066; JFM 51.0173.01
Full Text: