On Second Moment of Selberg Zeta-Function for σ=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma =1$$\end{document}

Let Z(s) be the Selberg zeta-function for the modular group. We consider the existence of the second moments of Z(s) and of its reciprocal on σ=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma =1$$\end{document}. The existence of such moments is related to the properties of certain Beurling natural numbers. Here the behavior of the counting function and the distribution of minimal gaps between these Beurling natural numbers are important. We also obtain unconditional upper bounds for these moments.