The Riemann zeta function on arithmetic progressions and denseness properties

In recent joint work with Sourmelidis, Steuding and Suriajaya, the author studied the distribution of values of the Riemann zeta function on arithmetic progressions in relation to denseness. We showed that one can find an infinitely long arithmetic progression such that the sequence of values of the Riemann zeta function on the right-half of the critical strip shifted with that arithmetic progression is contained in any given subset of the Riemann sphere. In this paper, we give an alternative proof which allows us to extend the result to the line 1+it\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+it$$\end{document} (t∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in {\mathbb {R}}$$\end{document}). We further show that we can relate the set of values of the Riemann zeta function on arithmetic progressions in the right-half of the critical strip to that on the critical line 1/2+it\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/2+it$$\end{document} (t∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in {\mathbb {R}}$$\end{document}). The latter gives a new approach to the question on denseness of ζ(1/2+it)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta (1/2+it)$$\end{document}.