On the periodic zeta-function (Q1873209)

Let \({\mathcal A} =\{a_m\}_{m=-\infty}^\infty\) be a periodic sequence of complex numbers with the period \(k>0\). The authors continue their work [Liet. Mat. Rink. 40, Spec. Iss., 28-32 (2000; Zbl 1024.11058)] and consider the periodic zeta-function, initially defined for \(\sigma = \Re s > 1\), \[ \zeta(s,{\mathcal A}) = \sum_{m=1}^\infty a_m m^{-s} = k^{-s}\sum_{q=1}^k a_q\zeta(s,q/k), \] where \(\zeta(s,\alpha)\) is the familiar Hurwitz zeta-function. In this paper they obtain some results on the value distribution of \(\zeta(s,{\mathcal A})\). Specifically, they obtain asymptotic formulas for the mean square integral \[ \int_1^T|\zeta(\sigma + it,{\mathcal A})|^2 \text{ d}t \] when \(\sigma = 1/2, 1/2 < \sigma < 1\) and when \(\sigma = 1\), which in view of the functional equation for \(\zeta(s,{\mathcal A})\) covers then essentially all the relevant cases. In addition, they obtain two results on the convergence of the probability measure \(P_T\), defined precisely in the text, involving the random element \(\zeta(s,\omega,{\mathcal A})\).