The number of lattice points below a logarithmic curve

For arbitrary real b > 1 and for a large real parameter t let R (b, t) be the number of lattice points between the curve by = x (1 ≤x≤t) and the x-axis, the points on the x-axis being counted with the factor 1 / 2. Furthermore, let the lattice rest\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ {\mit\Gamma} (b, t) $\end{document} be defined as the difference between R (b, t) and the area of the domain. The asymptotic behaviour of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ {\mit\Gamma} (b, t) $\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ t\rightarrow \infty $\end{document} and also for b→ 1+ are investigated.