On the periodic Hurwitz zeta-function with rational parameter

Abstract. The periodic Hurwitz zeta-function \(\zeta(s,\alpha;\mathfrak{a})\), \(s=\sigma+it\), is a generalization of the classical Hurwitz zeta-function, and is defined, for \(\sigma > 1\), by the series \(\zeta(s,\alpha;\mathfrak{a})=\sum\limits_{m=0}^{\infty}\frac{a_m}{(m+\alpha)^s}\), where \(\mathfrak{a}=\{a_m\}\) is a periodic sequence of complex numbers, and \(0 < \alpha\leqslant1\) is a fixed parameter. In the paper, theorems on the approximation of analytic functions by discrete shifts \(\zeta(s+ikh, \alpha; \mathfrak{a})\), \(k=0,1,\dots\), \(h>0\), with rational \(\alpha\) are obtained.