The mean-square of the Lerch zeta-function near the critical line (Q1881798)

For \(\sigma > 1\), the Lerch zeta-function with parameters \(\alpha\), \(0 < \alpha \leq 1\), \(\lambda\) real, is defined by \[ L(\lambda, \alpha, s) = \sum_{m=0}^\infty {{e^{2\pi i \lambda m}} \over {(m+\alpha)^s}}. \] It is continuable to the whole plane (except for at most a simple pole at \(s=1\)). The authors are interested in the mean square \[ \int_0^T | L(\lambda, \alpha, \sigma + it)| ^2\, dt, \] when \(\sigma\) is near \(1\over 2\), more precisely, when \[ \sigma = \sigma_T = {1\over 2} + {1 \over {\ell_T}}, \text{ where } 0 < \ell_T \to \infty. \] Using the approximate functional equation for the Lerch zeta-function [see \textit{R. Garunkštis, A. Laurinčikas} and \textit{J. Steuding}, Math. Notes 74, No. 4, 469--476 (2003); translation from Mat. Zametki 74, No. 4, 494--501 (2003; Zbl 1096.11034)], the authors prove: \[ \int_0^T | L(\lambda, \alpha, \sigma_T + it)| ^2\, dt = T \ell_T \left(1+ O_{\lambda, \alpha}\left( {1 \over {\ell_T}}\right) + O_{\lambda, \alpha} \left( \exp\left\{ - {{\log T} \over {\ell_T}} \right\}\right) \;\right), \] if \(\ell_T = o(\log T)\), \[ \int_0^T | L(\lambda, \alpha, \sigma + it)| ^2\, dt = T \log T \left( 1 + O_{\lambda, \alpha}\left({1\over{\log T}}\right) + O_{\lambda, \alpha} \left({{\log T}\over {\ell_T}}\right) \right), \] if \(\log T = o(\ell_T)\), and \[ \int_0^T | L(\lambda, \alpha, \sigma + it)| ^2\, dt = {1\over2} T \log T \left( 1- e^{- {1\over\kappa}}\right)^2 + o(T \log T), \] if \(\lim_{T\to\infty} {{\ell_T}\over {\log T}} = \kappa\).