The joint distribution of periodic zeta-functions (Q2898759)

Let \(\mathfrak a = \{a_m: m \in \mathbb N\}, \mathfrak b= \{b_m: m \in \mathbb N\}\) be multiplicative sequences of complex numbers with period \(k, \ell\) respectively. Define NEWLINE\[NEWLINE \zeta(s;\mathfrak a) =\sum_{m=1}^\infty \frac {a_m}{m^s} \text{ and } \zeta(s, \alpha, \mathfrak b)= \sum_{m=0}^\infty \frac{b_m}{(m+\alpha)^s}.NEWLINE\]NEWLINE The authors prove a joint approximation of given collection of analytic functions by collections of shifts of functions of the above type. This is applied to prove functional independence for these zeta-functions. The results extend work of \textit{S.~M.~Voronin} [Acta Arith. 27, 493--503 (1975; Zbl 0308.10025)], \textit{S.~M.~Gonek} [Analytic properties of zeta and L-functions, Ph.~D.\ Thesis, University of Michigan (1979)], and \textit{B.~Bagchi} [Math.~Z. 181, 319--334 (1982; Zbl 0479.10028)].