On second moment of Selberg zeta-function for \(\sigma =1\) (Q2047398)

For the Selberg zeta function \(Z(s)\) of the modular group, the authors first prove \(Z^{\pm1}(1+it)\ll\log T\) as \(T\to\infty\) and that both the second moment and its reciprocal on \(\Re(s)=1\) exist satisfying \[ \frac1T\int_1^T|Z(1+it)|^{\pm2}dt \ll\frac{\log^2T}{(\log\log T)^{2-\varepsilon}} \] for any \(\varepsilon>0\) as \(T\to\infty\). Then by using this theorem they deduce \[ \lim_{T\to\infty}\frac1{2T}\int_{-T}^T|Z(1+it)|^2dt =\sum_{n=1}^\infty\frac{a_n^2}{y_n^2} \] and that \[ \lim_{T\to\infty}\frac1{T}\int_{1}^T|Z(1+it)|^{-2}dt =\sum_{n=1}^\infty\frac{b_n^2}{y_n^2}, \] where the Dirichlet series expression of the Selberg zeta function and its reciprocal are denoted by \[ Z(s)=\sum_{n=1}^\infty\frac{a_n}{y_n^s}\quad(\Re(s)>1) \] and \[ Z^{-1}(s)=\sum_{n=1}^\infty\frac{b_n}{y_n^s}\quad(\Re(s)>1) \] with \(1=y_1<y_2<y_3\cdots\).