You can use combinations of \pagenumbering{<style>} \setcounter{page}{1} to adjust the setting of the page number (and reset it to 1 with every new style):
\documentclass[titlepage]{amsart}
\usepackage{lipsum}% Just for this example
\begin{document}
\pagenumbering{gobble}% No page numbering
\title{The title}
\author{The author}
\maketitle
\tableofcontents
\clearpage
\pagenumbering{roman}% roman page numbering
\section{Introduction} % the 0 section
\lipsum[1-50]
\clearpage
\pagenumbering{arabic}% Arabic page numbering
\section{First section} % the second section
\lipsum[1-50]
\end{document}
Note that this might cause problems if you're using hyperref, as you may have duplicate destinations with regards to the title and ToC pages. To keep hyperref happy, you could use the following setup:
\documentclass[titlepage]{amsart}
\usepackage[pdfpagelabels]{hyperref}
\usepackage{graphicx,etoolbox}
\usepackage{lipsum}% Just for this example
\begin{document}
\setcounter{page}{-1}
\title{The title}
\author{The author}
\begingroup
\makeatletter
\let\ps@firstpage\ps@empty
\maketitle
\patchcmd{\ps@headings}{\thepage}{}{}{}% Remove \thepage from \@evenhead
\patchcmd{\ps@headings}{\thepage}{}{}{}% Remove \thepage from \@oddhead
\pagestyle{headings}
\tableofcontents
\clearpage
\endgroup
\pagenumbering{roman}
\setcounter{page}{1}
\section{Introduction} % the 0 section
\lipsum[1-50]
\clearpage
\pagenumbering{arabic}
\setcounter{page}{1}
\section{First section} % the second section
\lipsum[1-50]
\end{document}
While \pagenumbering{gobble} will gobble \thepage from the visible page numbering, hyperref still uses \thepage for referencing purposes. So, we start the page numbering at a number that will naturally increase to i (roman) at the start of the first section; that is, -1 in this case, and just remove \thepage from the header.
\texttt{}with a non text does not make sense. – Sigur Feb 23 '15 at 19:04