Uniform Distribution Modulo 1 and the Joint Universality of Dirichlet L-functions

In the paper, we prove a joint universality theorem on the approximation of a collection of analytic functions by a collection of shifts of Dirichlet L-functions L(s + iτ,χ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \chi $$\end{document}j), where τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tau $$\end{document} takes values from the set {kα: k = 0, 1, 2, . . . } with 0 <α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha $$\end{document}< 1. The proof of this theorem uses the theory of uniform distribution modulo 1.