%\documentclass[12pt]{article}  % standard LaTeX, 12 point type 
\documentclass[12pt]{amsart} % AmS LaTeX, 12 point type
%\usepackage{amsfonts,latexsym} % for standard LaTeX
\usepackage{amsthm}
\usepackage{amssymb, amsmath}
\input{psfig.sty} % for imported eps files

% commands for sequential numbering, italicized text in theorems, etc.

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{definition}
\newtheorem{defn}{Definition}[section] 
\newtheorem{example}{Example}[section]

% unnumbered environments:

\theoremstyle{remark}
\newtheorem*{remark}{Remark}
\newtheorem*{notation}{Notation}
\newtheorem*{note}{Note}

\title{Proof of the Riemann Hypothesis}
\author{Julius Caesar}
\date{\today}         

\begin{document}

% examples of some useful macros:

\newcommand{\CC}{\mathbb C} % blackboard math , for ``complex,'' etc

%\newcommand{\qed}{$\Box$}      % box  indicating end of proof.

% for a sequence of unnumbered displayed equations:
\newcommand{\beas}{\begin{eqnarray*}} 
\newcommand{\eeas}{\end{eqnarray*}} 

\newcommand{\bm}[1]{{\mbox{\boldmath $#1$}}} % for boldface math symbols 

% for binomial coefficients (AmS-LaTeX)
\newcommand{\bc}[2]{\genfrac{(}{)}{0pt}{}{#1}{#2}}

% for two rows, say under a summation sign (AmS-LaTeX)
\newcommand{\tworow}[2]{\genfrac{}{}{0pt}{}{#1}{#2}}

\begin{abstract}
put abstract here

% An abstract is a short summary of the main points in the project.  It
% does not mention background material or detail.

\end{abstract}

\maketitle


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Introduction}\label{sec:intro}

Your introduction section should provide an outline of your project.
Include the motivation, methods used, and your conclusions.  Keep
technical information out of your introduction.

To conclude your introduction, give a road map describing
the sections in the rest of your document.  This road map provides a
quick reference for the reader interested in a particular topic or
aspect of your project.

You must use use \LaTeX\ to format your document.  \LaTeX\ is the
typesetting system universally used to communicate mathematics. It
\LaTeX\ provides the best and (after you get used to it) the easiest
way to typeset all manner of mathematical expressions.  Learning
\LaTeX\ is an important part of your mathematical education because it
is the standard means of producing math papers and books.  Wherever
possible, use the \AmS-\LaTeX\ package commands.  They are
specifically designed for mathematics as opposed to other science
disciplines.  The \AmS-\LaTeX\ packages are especially useful for
symbols and typesetting equations: the standard \LaTeX\ commands for
equations just aren't as good.

One free and particularly good reference for standard \LaTeX\ is given
at \cite{oetiker}.  A short reference for \AmS-\LaTeX\ is given at
\cite{downes}.  The course website has links to these sources. 
The standard reference book for \LaTeX, which includes
a section on \AmS-\LaTeX, is \cite{goossens}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage

\section{\LaTeX\ Examples}\label{sec_eg}

Here are a few examples of \LaTeX\ goodies. You should view the
{\tt .tex} file as you read this section.   

\LaTeX\ will indent after a blank line unless told not to do so.

Some symbols, such as the dollar sign and the percent sign, have
special meanings in a \LaTeX\ document.  To tell the computer that
want such a symbol printed in the text instead of interpreted as an
instruction to the computer precede the symbol by a backslash:
\verb+\$+, \verb+\%+.  A few symbols have special commands, e.g.,
\verb+\backslash+ for a backslash. You can also produce a backslash
with the verbatim environment, but most likely you will have no need
for this feature. 

The percent sign, \% when used without the backslash, tells the
computer to ignore the rest of that line.   %This sentence will not appear.

There are many ways to write formulas in \LaTeX.  Inline formulas are
deliminated by dollar signs \$.     %without the backslash  
For example, $z^2 = \sin \left(\theta_0 \right)$, or $2 \times 2 \neq 5$.

Here are two ways to display equations without a reference number:

$$\int_1^x \frac{dt}{t} = \ln x.$$

\[
\int_1^x \frac{dt}{t} = \ln x.
\]

An equation with a number for later referencing:

\begin{equation}\label{eq:limit}
\lim_{n \to \infty}\, \sum_{k = 1}^n \frac{1}{k^2} = \frac{\pi}{6}.
\end{equation} %

A reference to the above equation: (\ref{eq:limit})

To group several unnumbered equations, aligning on the $=$ (or some
other) sign, use the \verb+\begin{eqnarray*}+ command.  Here the
macros \verb+\beas+ and \verb+\eeas+ have been set up at the
beginning of the file. The computer aligns at the \&:
%
\beas
x_1 + 2x_2 + 3x_3 &=& 7 \\ &>& 5\\ y &=& mx + c.
\eeas
%
To number only the middle of these three equations, use
\verb+\begin{eqnarray}+ and \verb+\nonumber+ as appropriate: 

%
\begin{eqnarray}
x_1 + 2x_2 + 3x_3 &=& 7 \nonumber\\ &>& 5\\ y &=& mx + c. \nonumber
\end{eqnarray}
%
For more examples of equation alignments, see \cite{downes}.

Use of boldface in math mode (using the \verb+\bm+ macro defined
above): 
  $$ \bm{\alpha+\beta=\frac 12}. $$

A matrix with all entries centered:
$$ \left[ \begin{array}{ccc} 1 & 2 & 3\\ 4 & 5 & \frac{37}{\pi} 
    \end{array} \right]
$$

A ``two-option'' formula:
  $$ |x| = \left\{ \begin{array}{rl} x, & \mathrm{if}\ 
         x\geq 0\\
         -x, & \mathrm{if}\ x\leq 0. \end{array} \right.
  $$
This example has several instructive features: 
\begin{itemize}\item The use of \verb+\left\{+, not \verb+\left{+ 
  \item  The use of \verb+{rl}+ to have the
first field right-justified and the second left-justified. 
  \item The use of \verb+\+ followed by a space to create a space in
  math mode. 
  \item The use of \verb+\mathrm+ to create roman font in math mode,
    which is necessary for text.
  \item The use of \verb+\right.+ to close the delimiter
    \verb+\left\{+. Every use of \verb+\left+ must be associated 
   with a later use of \verb+\right+. If no right delimiter is wanted,
   use \verb+\right.+.
\end{itemize}


Writing a binomial coefficient in \AmS-\LaTeX\ using the macro
\verb+\bc+:
   $$ \bc{a+b}{3} $$

Creating two rows under a summation sign in \AmS-\LaTeX using the
macro \verb+\tworow+:
  $$ \sum_{\tworow{n\geq 0}{n\equiv 2\,(\mathrm{mod}\,3)}} n^2x^n $$
Note the method for writing a congruence. The standard \LaTeX\ command
\verb+\pmod+ does not produce as nice spacing. If you use a lot of
congruences, you will want to write a macro for
\verb+\,(\mathrm{mod}\,#1)+, where \verb+#1+ indicates the postition
of the object of the macro (as for the macro \verb+\bm+ at the
beginning of this document). 

Be sure to use large parentheses (or other delimiters) when necessary:
  $$ \mathrm{not}\ (\frac{x^2+1}{\sqrt{e^{x^2}+1}}),\ \mathrm{but\
    rather}\ \left(\frac{x^2+1}{\sqrt{e^{x^2}+1}}\right).
  $$

An example of a table used as a figure is given by
Figure~\ref{fig:animals}. 

\begin{figure}
\begin{center}
\begin{tabular}{r||c|c|l}
animal & legs & size & Class\\ \hline
millipede & lots & tiny & Arthropoda\\
platypus & 4 & small & Mammalia\\
shark & 0 & large & Chondrichthyes\\
 \end{tabular}
\end{center}
\caption{Animals}
\label{fig:animals}
\end{figure}
%line \begin{tabular}{r||c|c|l} line tells the computer: three columns
% || inserst a double vertical line
% |  inserts a vertical line
% r  justifies the first column on the right
% l  justifies the fourth column on the left
% c  centers the second and third columns
% &  between columns
% \\ at the end of a column
% \hline inserts a horizontal line
% no * was used, so the table gets numbered

Importation of the encapusulated PostScript file foo.eps into a
figure is shown below. The second line needs to be uncommented (i.e.,
remove \% at the beginning). It is included here since otherwise the
computer will stop compiling when it doesn't find foo.eps.   

\begin{figure}\label{myfirstfig}
%\centerline{\psfig{figure=foo.eps}}
replace this line with uncommented previous line
\caption{Phooey}
\label{fig:foo}
\end{figure}

\bigskip
An example of a ``conjecture environment'' is given below, in
Conjecture~\ref{myfirstconj}. Theorems, lemmas, propositions,
definitions, etc., all use the same command with the 
appropriate name changed.  The block of commands at the top
of this {\tt .tex} file causes them to be numbered sequentially within
sections, which is the preferred choice.

\begin{conjecture}[originator unknown]\label{myfirstconj}
Define $x_0=2$ and $x_n=x_{n-1}-\frac{1}{x_{n-1}}$ for $n\geq 1$. Then
the sequence $x_0, x_1, \dots$ is unbounded.
\end{conjecture}

An example of the remark environment. Since \verb+*+ was placed after
\verb+\newtheorem+ in defining the environment, it isn't
numbered. Since the line \verb+\newtheorem*{remark}{Remark}+ is placed
under \verb+\theoremstyle{remark}+, the font style is different from
theorems, conjectures, etc.

\begin{remark}
This is a remark.
\end{remark} 

Be sure to use \verb+\proof+ and \verb+\qed+ to begin and end proofs:

\proof This is a proof. \qed

\emph{Citations.} Use the bibliography environment as shown at the end
of this paper.  See the bibliography below for referencing a website
\cite{downes}\cite{oetiker}, a book \cite{goossens}, a journal paper
\cite{a-d}, and a paper within a book \cite{b-r}. There are many
different styles required by different publishers. The one given below
is typical. To cite a certain part of a reference, such as Theorem~5
of reference \cite{a-d}, use \verb+\cite[Thm.~5]{a-d}+. (The tilde
preceding the 5 prevents a possible linebreak after Thm., which would
look bad.) All references in the bibliography should be cited in the
text.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Filetype translations}\label{sec_translations}


Produce a {\tt .tex} file using any text editor that produces ASCII files.
Tex files conventionally bear the extension {\tt .tex}. 
The command 

\centerline
{\tt latex filename.tex}
\noindent
produces either error reports or a file {\tt filename.dvi}. 
Many systems have screen {\tt .dvi} viewers. Under Linux for example
you can use {\tt xdvi filename}. Most printers print postscript files.
Under Linux, files can be converted from {\tt .dvi} to {\tt .ps} using 

\centerline
{{\tt dvips -f filename.dvi > filename.ps}.} 
\noindent
and from {\tt .ps} to {\tt .pdf} using

\centerline
{{\tt ps2pdf filename.ps}.} 
\noindent
If the file does not contain {\tt .eps} elements,
{\tt .pdf} output can be obtained directly using 

\centerline
{{\tt pdflatex filename.tex}.}

Matlab can be made to produce {\tt .eps} files by typing 

\centerline
{\tt print -deps filename}
\noindent
at the prompt.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{9}  
% ``9'' specifies that the bibliography
% has just one-digit reference numbers.
  
\bibitem{a-d} D. J. Aldous and P. Diaconis, Hammersley's
  interacting particle process and longest increasing subsequences,
  \emph{Probab.\ Th.\ Rel.\ Fields} \textbf{103} (1995), 199--213.

\bibitem{b-r} J. Baik and E. Rains, Symmetrized random
permutations. In \emph{Random Matrix Models and Their Applications},
Math.\ Sci.\ Res.\ Inst.\ Publ.\ \textbf{40}, Cambridge University
Press, 2001, pp.~1--19.

\bibitem{downes}
M. Downes, ``Short Math Guide to \LaTeX,'' \\
\verb+http://www.ams.org/tex/short-math-guide.html+

\bibitem{goossens}
M. Goossens, F. Mittelbach, and A. Samarin, \emph{The \LaTeX\
Companion}, Addison--Wesley, Reading, MA, 1994.

\bibitem{oetiker}
T. Oetiker et al.,  ``The Not So Short Introduction to \LaTeXe,'' \\
\verb+http://www.ctan.org/tex-archive/info/lshort/english/lshort.pdf+

\end{thebibliography}

\end{document}