On the periodic zeta-function. II (Q1873229)

The authors continue their work [Lith. Math. J. 41, No. 2, 168--177 (2001); translation from Liet. Mat. Rink. 41, No. 2, 214--226 (2001; Zbl 1025.11028)] on the periodic zeta-function \(\zeta(s, \mathcal A)=\sum_{m=1}^\infty a_mm^{-s}\) \((\Re s>1)\), where \(\{a_m\}\) is a sequence of complex numbers with period \(k\) \((\geq 1)\). In the first part of this paper they established an asymptotic formula (\(1/2 < \sigma < 1\) fixed) for \[ (1) \int_1^T|\zeta(\sigma + it,{\mathcal A})|^2 \text{ d}t. \] In the present work they sharpen their result on (1) by using a sharp approximate functional equation for \(\zeta(s, \mathcal A)\), which is the analogue of the classical Riemann-Siegel formula for the Riemann zeta-function \(\zeta(s)\). The proof, based on complex integration technique, is similar to the one for \(\zeta(s)\).