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\begin{document}
\title{\bf{Proposal}}
\author{Jos\'{e} Gim\'{e}nez}
\date{June 23, 2005}
\maketitle

\noindent
{\setlength{\baselineskip}%
{1.5\baselineskip} The unrestricted partition function $p(n)$
counts the number of ways a positive integer $n$ can be expressed
as a sum of positive integers $\leq n$. The partition function is
generated by Euler's infinite product
\begin{equation} \label{eq:infprod}
F(x)=\prod_{m=1}^\infty (1-x^m)^{-1}=1+\sum_{n=1}^\infty p(n) x^n
\qquad |x|<1
\end{equation}
This establishes a relationship between the Dedekin Eta function defined as
\begin{equation} \label{eq:eta}
\eta(\tau)=e^{\frac{\pi i \tau}{12}}\prod_{n=1}^\infty (1-e^{2\pi
i n\tau}) \qquad \tau \in H
\end{equation}
and the partition function $p(n)$, given by
\begin{equation} \label{eq:etafourier}
\eta^{-1}(\tau)=\sum_{m=-1}^\infty p(m+1)e^{2\pi
i(m+\frac{23}{24})\tau} \qquad \tau \in H
\end{equation}

Since $\eta(\tau)$ is a modular form of weight ${1/2}$,
$\eta^{-1}(\tau)$ has weight $-{1/2}$, and the fact that
$\eta^{-1}(\tau)$ is meromorphic and (\ref {eq:etafourier}) makes
$\eta^{-1}(\tau)$ a modular form of negative weight. From (\ref
{eq:infprod}) we can see that
\begin{equation}\label{eq:pn}
p(n)=\frac{1}{2\pi i}\int_{C}\frac{F(x)}{x^{n+1}}dx
\end{equation}
where $C$ is a positively oriented simple closed curve on the unit
circle containing zero  in its interior.

The circle method consist of making the change of variable $x=e^{2
\pi i \tau}$ for every $N$ and choose  a path of integration
$P(N)$ Joining $i$ and $i+1$ through the upper arcs of  the Ford
circles of the Farey series $F_N$. An then from (\ref {eq:pn}) we
get

\begin{equation}\label{eq:pn2}
p(n)=\int_{i}^{i+1}F(e^{2 \pi i \tau}) e^{-2 \pi i n \tau}d \tau =
\int_{P(N)}F(e^{2 \pi i \tau}) e^{-2 \pi i n \tau}d \tau
\end{equation}



Rademacher [9] modified the circle method first introduced by
Hardy and Ramanujan  in order to get an exact formula for the
partition function
\begin{equation}
p(n)=\frac{1}{\pi \sqrt{2}}\sum_{k=1}^{\infty} A_k (n) \sqrt{k}
\frac{d}{dn}\left( \frac{\sinh
\left\{\frac{\pi}{k}\sqrt{\frac{2}{3} \left( n -
\frac{1}{24}\right)} \right\}}{\sqrt{  n - \frac{1}{24}}}\right)
\end{equation}
where
\begin{equation}
A_k (n)= \sum_{\begin{array}{c}
  0 \leq h < k \\
  (h,k)=1 \\
\end{array}}e^{\pi i s(h,k) - 2 \pi i n h /k}
\end{equation}

 On the other hand, Knopp and Mason [3] obtained growth conditions of the
Fourier coefficients of vector-valued modular forms. In [4] they
developed a general theory of vector-valued modular forms. The
following definition is given in [4]: Let
$F(\tau)=(F_1(\tau),\ldots,F_p(\tau))$ be a $p$-tuple of functions
holomorphic in the complex upper half-plane $H$ and $\rho:\Gamma
\longrightarrow GL(p,C)$ a $p$-dimensional complex
representation.$(F,\rho)$, or simply $F$, is a vector-valued form
of real weight $k$ on the modular group $\Gamma=SL(2,Z)$ if

\begin{enumerate}
    \item For all $V=\left(%
\begin{array}{cc}
  a & b \\
  c & d \\
\end{array}%
\right) \in \Gamma$ we have

    \begin{equation} \label{eq:ftau}
    (F_1(\tau),\ldots,F_p(\tau))^t\mid_k
    V(\tau)=\rho(V)(F_1(\tau),\ldots,F_p(\tau))^t
    \end{equation}
    \item Each component function $F_j(\tau)$ has a convergent
    $q$-expansion meromorphic at infinity:
    \begin{equation} \label{eq:fourier}
    F_j(\tau)=\sum_{n\geq h_j} a_n(j)q^{\frac{n}{N_j}}
    \end{equation}
    with $N_j$ a positive integer and $q=e^{2\pi i \tau}$. The
    Slash operator $\mid_k V$ in (\ref {eq:ftau}) is defined by:
        \begin{equation} \label{eq:slash}
   F\mid_k V(\tau)=F\mid ^v _k V(\tau)=v(V)^{-1}(c\tau +
   d)^{-k}F(V\tau).
    \end{equation}
\end{enumerate}
The goal of my thesis is to:
\begin{itemize}
   \item
    Apply the circle method to vector-valued modular forms of negative weight in order to get the exact formula for the Fourier coefficients.
    \item
    Show that exists a vector-valued modular form of negative weight.
    \item
     Find an upper and lower bound in the dimension of $M(k,\rho)$, the space of vector-valued modular forms when $k$ is negative.
\end{itemize}
\newpage
\begin{thebibliography}{99}
    \bibitem{[1]}
M. ~Knopp, Modular Functions in Analytic Number Theory, Markham
Publishing, Illinois, 1970.
    \bibitem{[2]}
M. ~Knopp and G. Mason, Generalized modular forms, J. Number
Theory 99 (2003), 1-28.
    \bibitem{[3]}
M. ~Knopp and G. Mason, On vector-valued modular forms and their
Fourier coefficients, Acta Arith. 110 (2003), no. 2, 117124.
    \bibitem{[4]}
M. ~Knopp and G. Mason, Vector-Valued Modular forms and Poincare
Series, Illinois Journal of Mathematics 48 (2004), 1345-1366.
    \bibitem{[5]}
H. ~Rademacher, On the partition function $p(n)$. Proc. London
Math. Soc. 43 (1937), 241-254.
    \bibitem{[6]}
H. ~Rademacher, A convergent series for the partition function
$p(n)$. Proc. Nat. Ac. Sci USA 23 (1937), 78-84.
    \bibitem{[7]}
H. ~Rademacher, The Fourier coefficients of the modular invariant
$J(\tau)$. Am. J. Math. 60 (1938), 501-512.
    \bibitem{[8]}
H. ~Rademacher,  The Fourier series and the functional equation of
the absolute modular invariant $J(\tau)$. Am. J. Math. 61 (1939),
237-248.
    \bibitem{[9]}
H. ~Rademacher, On the expansion of the partition function in a
series. Ann. Math. 44 (1943), 416-422.
\end{thebibliography}
\end{document}