Functional independence of periodic Hurwitz zeta functions
Given a periodic sequence \({\mathfrak a}:= \{a_m\mid m\in\mathbb Z, m\geq 0\}\) of complex numbers \(a_m\), let \[ \zeta(s,\alpha;{\mathfrak a})= \sum^\infty_{m=0} a_m(m+ \alpha)^{-s}. \] The so-called periodic Hurwitz zeta-function \(s\mapsto \zeta(s,\alpha;{\mathfrak a})\) can be analytically continued to the whole complex plane. Making use of an universality theorem of J. Steuding, the author proves functional independence and joint functional independence for a set of periodic Hurwitz zeta-functions with algebraically independent parameters. To state the simplest of his results, let \(F_j:\mathbb C^N\to \mathbb C\), \(0\leq j\leq n\), be a continuous function and suppose that \(\alpha\) is a transcendental number. If \[ \sum^n_{j=0} s^j F_j(y(s), y^{(1)}(s),\dots, y^{(N-1)}(s))= 0 \] for every \(s\) in \(\mathbb C\), where \(y(s):= \zeta(s,\alpha; {\mathfrak a})\), then \(F_j(\mathbb C^N)= \{0\}\) for \(0\leq j\leq n\).
scientific article; zbMATH DE number 5352318
scientific article