\mychapter{Ramanujan's Mock Theta Functions}
In this chapter, we prove Theorem \ref{mainthm3} and Corollary \ref{question}, which are joint work with Michael Griffin and Ken Ono in \cite{Griffin-Ono-Rolen}. We then prove Theorem \ref{suppthm3}. We begin by stating the following result, which follows from the work of Zwegers \cite{Zwegers1,Zwegers2}.

\begin{fact}\label{lacunary} Suppose that $M(z)$ is one of Ramanujan's alleged examples of a mock theta function.
Thanks to Zwegers \cite{Zwegers1,Zwegers2}, there are
 integers $\gamma$ and $\delta$ for which $q^{\gamma}M(\delta z)=:f^{+}(z)$ is the holomorphic
 part of a weight 1/2  harmonic weak Maass form $f(z)$ on a congruence subgroup $\Gamma_1(N)$.
 Moreover, the nonholomorphic part of this form is the period integral of a weight
  3/2 unary theta function.  In particular, there are finitely many positive integers $\delta_1,\dots,
 \delta_s$ for which
 $c_f^{-}(n)=0$
 unless $n= -\delta_i m^2$ for some $1\leq i\leq s$ and some integer $m$.
 \end{fact} 
 The rest of the chapter is organized as follows.
In $\S\ref{PoincareSec}$ we describe the construction of the Poincar\'e series. In $\S\ref{ProofMainThm3}$ we give with the proof of Theorem \ref{mainthm3} and Corollary \ref{question}. We conclude with the proof of Theorem \ref{suppthm3} in $\S\ref{ProofSuppThm3}$.
 \section{Poincar\'e Series}\label{PoincareSec}
We require
Maass-Poincar\'e series, which were considered previously in work of Niebur \cite{Niebur1, Niebur2}.
Their principal parts will serve as a basis for the principal parts of the mock theta functions. For $s\in\C$ and $y\in\R-\{0\}$ we let $\mathcal{M}_s(y):=|y|^{-\frac k2}M_{\frac k2\operatorname{sgn}(y),s-\frac12}(|y|)$,
where $M_{\nu,\mu}$ is the usual $M$-Whittaker function which satisfies
\[\frac{\partial^2u}{\partial z^2}+\left(-\frac14+\frac{\nu}{z}+\frac{\frac14-\mu^2}{z^2}\right)u=0.\]

Since spaces of forms on $\Gamma_1(N)$ are a direct sum over the spaces of Maass forms on $\Gamma_0(N)$ with Nebentypus, it suffices to construct Poincar\'e series on $\Gamma_0(N)$ with arbitrary Nebentypus $\chi$.
For a positive integer $m$, we define
$\phi_{-m,s}(z):=\mathcal{M}_s(-4\pi my)e(-mx)$,
and we define the Poincar\'e series on $\Gamma_0(N)$ with Nebentypus $\chi$ and weight $k\in\frac12+\Z$ by

\begin{equation}\label{PoinDef}\mathcal{F}_k(-m,s,z):=
\displaystyle\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma_0(N)}\left(\frac{c}{d}\right)^{-2k} \epsilon_d^{2k}\chi(d)^{-1}(\phi_{-m,s}|_k\gamma)(z).\end{equation}
It turns out that $\phi_{-m,s}(z)$ is an eigenfunction of $\Delta_k$ with eigenvalue
$s(1-s)+(k^2-2k)/4$.
Therefore $\mathcal{F}_k(-m,s,z)$ is a weak Maass form of weight $k$ on $\Gamma$ with character $\chi$ whenever the series is absolutely convergent. This is clear if $\Re(s)>1$ as $\phi_{-m,s}(z)=O(y^{\Re(s)-\frac k2})$ as $y\rightarrow0$. To obtain a \emph{harmonic} Maass form, we choose $s=\frac{k}{2}$ (or $s=1-\frac{k}{2}$ if $k<1$). Convergence for this choice of $s$ for weight
 $k\in \frac{1}{2}+\Z$ Poincar\'e series is only questionable if $k=1/2$ or $k=3/2$. We are primarily interested in the case when $k=1/2$.

% We now give exact formulas for the coefficients of these series from which we easily %deduce asymptotics. To do so, we will require the function
%$\mathcal W_s(y):=|y|^{-\frac {k}{2}}W_{\frac {k}{2}\operatorname{sgn}(y),\, %s-\frac12}(|y|)$,
%where $W_{\nu,\mu}$ is the $W$-Whittaker function.
% We also recall the relations
%$\mathcal{M}_{\frac k2}(-y)=e^{\frac y2}$,
%$\mathcal{W}_{1-\frac k2}(y)=\mathcal{W}_{\frac k2}(y)=e^{-\frac y2}$,
%and
%$\mathcal{W}_{1-\frac k2}(-y)=\mathcal{W}_{\frac k2}(-y)=e^{\frac y2}\Gamma(1-k,y),$
%where $\Gamma(a,x):=\int_x^{\infty}e^{-t}t^a\frac{\mathrm dt}{t}$
%is the incomplete Gamma function.


%Now to compute the Fourier expansion, we describe a set of representatives for %$\Gamma_{\infty}\backslash\Gamma_0(N)$. For each pair %$(c,d)\in\mathbb{N}\times\mathbb{Z}$ such that $N|c$ and $(c,d)=1$ we choose a matrix %$\left(\bigl( \begin{smallmatrix}a&b\\ %c&d\end{smallmatrix} \bigr)\right)\in\operatorname{SL}_2(\mathbb{Z})$. In addition, we have the %identity matrix.
% The contribution from the identity matrix gives the term $q^{-m}$. For the terms with %$c>0$ we see that the other terms contribute the quantity



%\begin{equation}
%\begin{split}
%\displaystyle\sum_{c>0,\, N|c}c^{-\frac12}\sum_{d\text{     (mod }c)^*}\left(\frac %cd\right)^{-2k}\epsilon_d^{2k}\chi(d)&\sum_{n\in\mathbb{Z}}(z+d/c+n)^{-\frac12}\\
%&\times\mathcal{M}_{s}\left(-\frac{4\pi m %y}{c^2|z+d/c+n|^2}\right)e\left(\frac{m}{c^2}\Re\left(\frac{1}{z+d/c+n}\right)\right).
%\end{split}
%\end{equation}
The Fourier expansion of such series is well known (for example, see
\cite{Niebur1,Fay,BruinierBorcherds, BringmannOno06,BringmannOnoMathAnnalen}).
We recall the \emph{Kloosterman sum} of weight $k\in\frac12+\Z$ for $\Gamma_0(N)$ with Nebentypus $\chi$.
\begin{equation} K_k(m,n,c,\chi):=\displaystyle\sum_{d\text{ (mod }c)^{\times}}\left(\frac cd\right)^{-2k}\epsilon_d^{2k}\chi(d)e\left(\frac{m\overline{d}+nd}{c}\right),
\end{equation}
where $d$ runs through primitive residue classes mod $c$ and $\overline{d}$ is the multiplicative inverse of $d$ mod $c$. We then have the following.


\begin{proposition}\label{PoincareFS}
If $m$ is a positive integer, then the Poincar\'e  series \\$\mathcal{F}_k(-m,z,s)$ for $\Gamma_0(N)$ with Nebentypus $\chi$  has the Fourier expansion
\[\mathcal{F}_k(-m,z,s) = \mathcal{M}_s(-4\pi my)e(-mx)+\sum_{n\in \Z}c(n,y,s)e(nx),\]
where the coefficients $c(n,y,s)$ are given by
\begin{equation}\footnotesize\left\{
\begin{array}{lr}
\frac{2 \pi i^{-k}\Gamma(2s)}{\Gamma(s-k/2)}\left|\frac{n}{m}\right|^{\frac{k-1}{2}}\displaystyle\sum_{c>0,N|c}
\frac{K_k(-m,n,c,\chi)}{c}J_{2s-1}\left(\frac{4\pi\sqrt{|mn|}}{c}\right)\mathcal{W}_s(4\pi ny),&n<0\\

\frac{2 \pi i^{-k}\Gamma(2s)}{\Gamma(s+k/2)}\left|\frac{n}{m}\right|^{\frac{k-1}{2}}
\displaystyle\sum_{c>0,N|c}\frac{K_k(-m,n,c,\chi)}{c}I_{2s-1}
\left(\frac{4\pi\sqrt{|mn|}}{c}\right)\mathcal{W}_s(4\pi ny),&n>0\\

\frac{4^{1-k/2} \pi^{1+s-k/2} i^{-k}|m|^{s-k/2}y^{1-s-k/2}\Gamma(2s-1)}{\Gamma(s+k/2)\Gamma(s-k/2)}
\displaystyle\sum_{c>0,N|c}\frac{K_k(-m,0,c,\chi)}{c^{2s}},&n=0
\end{array}
\right.\end{equation}
\end{proposition}


 In the proposition above, $I_k$ is the usual modified Bessel function and $J_k$ is the Bessel function of the first kind. If $s\geq1$ and equals $k/2$ or $1-k/2$, then these Poincar\'e series converge and are harmonic weak Maass forms. For $k=1/2$
 it is known that the formulas still hold. For completeness, we shall give
  brief remarks below concerning the convergence.


Before we discuss the weight $1/2$ case, we stress that this
proposition allows us to easily determine the asymptotics of the coefficients
of holomorphic parts of harmonic weak Maass forms.
This follows from the well-known asymptotic
 \begin{equation}\label{BesselAsymp}
I_k(x)\sim\frac{e^x}{\sqrt{2\pi x}}\left(1-\tfrac{4k^2-1}{8x}+\ldots\right).
\end{equation}
The Poincar\'e series constructed above have nonconstant principal parts only at
the cusp infinity.
 We may similarly construct Poincar\'e series at any cusp $h$. We let $\mathcal{F}_{k}(-m,s,z,h)$ denote the Poincar\'e series which is defined by modifying (\ref{PoinDef}) as
\[\mathcal{F}_k(-m,s,z,h)
:=\displaystyle\sum_{\gamma\in\Gamma_{h}\backslash\Gamma_0(N)}\left(\frac{c}{d}\right)^{-2k} \epsilon_d^{2k}
\chi(d)^{-1}(\phi_{-m,s}|_k\gamma)(z),\]
where $\Gamma_h$ is the stabilizer of $h$. As in the case of the cusp at infinity, we obtain a weak Maass form with order $-m$ principal part at the cusp $h$ and constant principal parts at all other cusps.

These facts allow us to conclude with the following crucial fact.

\begin{fact}\label{nonlacunarity}
Suppose that $f(z)$ is a weight 1/2 harmonic weak  Maass form with a
nonconstant principal part at some cusp.
Let $f_P(z)$ be the weight 1/2 harmonic weak Maass form that is a linear combination
of Maass-Poincar\'e series which matches, up to constants, the principal parts of $f(z)$ at all cusps. By Theorem~\ref{mainthm3}, it follows that $f(z)-f_P(z)$ is a weight 1/2 holomorphic modular form,
which, by the Serre-Stark Basis Theorem (for example, see \cite{cbms}), implies that $f(z)-f_P(z)$ is a linear combination of weight 1/2 unary
theta functions. Therefore,
the
subexponential growth of the $I$-Bessel function, combined with the periodicity of the Kloosterman sums in $n$,
when $m$ and $c$ are fixed,
then implies that a positive
proportion of the coefficients of the holomorphic part of $f^+(z)$
are nonzero. Indeed, this gives arithmetic progressions of coefficients with {\it smooth} asymptotic subexponential growth.
\end{fact}



\begin{remark}
We briefly discuss the convergence in Proposition \ref{PoincareFS}
for weight 1/2 harmonic weak Maass forms.
To show this, we need similar estimates for sums of the Kloosterman sums as in Theorem 4.1 of \cite{BringmannOno06}.  In that work  the Kloosterman sums were rewritten as Salie-type sums, which were then estimated using the equidistibution of CM-points. It is clear that the shape of the Salie-type sums do not depend on the multiplier system in a crucial way. Alternatively, the more general case, results of Goldfeld and Sarnak in \cite{Goldfeld-Sarnak} and the spectral theory of automorphic forms apply.  By the asymptotics for Bessel functions, it suffices to consider the continuation of the Selberg-Kloosterman zeta function
\begin{equation}
Z_{n,m}(s,\chi):=\displaystyle\sum_{c>0}\frac{K_k(-m,n,c,\chi)}{c^{2s}}.
\end{equation}
Namely, for $k=1/2$ we need to show convergence at $s=3/4$. The convergence we require was shown for a special case in Theorem 2.1 of \cite{Folsom-Ono}. The general case follows \emph{mutatis mutandis}. 
\end{remark}
\begin{theorem}%[2.1 of \cite{Folsom-Ono}]
If $m$ is a positive integer, then $Z_{n,m}(s,\chi)$ is convergent at $s=3/4$.
\end{theorem}


% The idea of the proof is as follows. In \cite{Goldfeld-Sarnak}, it is shown that for a %certain class of Poincar\'e series that the Selberg Kloosterman zeta function is %essentially a Petersson inner product of two Poincar\'e series when $m,n,k>0$. Realizing %the zeta function in this form allows one to obtain an analytic continuation for all %$s>1/2$. This continuation has simple poles whose location and residues are determined by %the spectral theory of automorphic forms with respect to the operator
%\[\widetilde{\Delta_k}=y^2\left( \frac{\partial^2}{\partial %x^2}+\frac{\partial^2}{\partial y^2} \right) -iky \frac{\partial}{\partial x}. \]
%By a remark of Pribitkin \cite{Pribitkin}, when $n<0$, $m>0$ one may derive a similar %residue formula by considering a different inner product of Poincar\'e series. For the %case $n>0$, $m<0$, one may use the case $n<0$, $m>0$ and the fact, for $k\in (0,1)$, %that
%$\overline{Z_{n,m}(\overline{s},\chi)}=Z_{-n-1,-m-1}(s,\overline{\chi})$. Analyzing this %residue formula for weight $1/2$ forms gives holomorphicity. Convergence then follows %using standard Perron-type formulas to relate series expansions for the zeta function to %their integral representations.



.



 \section{The Proof of Theorem~\ref{mainthm3} and Corollary~\ref{question}}\label{ProofMainThm3}
Here we prove Theorem~\ref{mainthm3} and Corollary~\ref{question}.

 \subsection{Proof of Theorem~\ref{mainthm3}}
 Suppose that $g(z)$ is a weakly holomorphic modular form on $\Gamma_1(N')$, for
  some $N'$, which cuts out the exponential singularities of $f(z)$ as $q$ approaches roots of unity.  Then $h(z):=f(z)-g(z)$ is a harmonic weak Maass form of weight $k$ on $\Gamma_1(\lcm(N,N'))$ with nonconstant nonholomorphic part.
  By Theorem~\ref{PrincipalPart}, $h(z)$ has a nonconstant principal
  part at some cusp.
Since the nonholomorphic part $f^{-}(z)$ exhibits exponential decay at cusps, it follows that
$h(z)$ is also $O(1)$ as cusps.
Suppose that $h(z)$ has a nonconstant principal part at infinity (a similar argument applies at other cusps). By choosing matrices $\left(\bigl( \begin{smallmatrix}a&b\\c&d\end{smallmatrix} \bigr)\right)\in \Gamma_1(\lcm(N,N'))$, combined with the fact that
$$\lim_{z\rightarrow i\infty}h\left(\tfrac{az+b}{cz+d}\right)=
\lim_{z\rightarrow i \infty}(cz+d)^k h(z),
$$
we find that infinitely many roots of unity are
exponential singularities for $h(z)$.


 \subsection{Proof of Corollary~\ref{question}}
 Suppose that $M(z)$ is one of Ramanujan's alleged examples of a mock theta function.
 Then there are integers $\gamma$ and $\delta$ for which $q^{\gamma}M(\delta z)=:f^{+}(z)$
 is the holomorphic part of the weight 1/2 harmonic weak Maass form.
 Now suppose that $g(z)$ is a weakly holomorphic modular form of some weight $k$ which
 cuts out the exponential singularities of $f(z)$.
Following the proof of Theorem 1.4 of \cite{BringmannOnoAnnals},
we can use Fact~\ref{lacunary}, Fact~\ref{nonlacunarity}, and the theory of quadratic (and trivial) twists to obtain a weight 1/2 weakly holomorphic modular form $\widehat{f}(z)$.
By Fact~\ref{nonlacunarity}, this can be done so that $\widehat{f}(z)$ is nontrivial and
has nonconstant principal parts at some cusp.
Applying the same procedure to $g(z)$ gives a weakly holomorphic modular form
$\widehat{g}(z)$. We then have that $\widehat{f}(z)$ and $\widehat{g}(z)$
cut out exactly the same exponential singularities at all roots of unity. By the discussion after
Corollary~\ref{question}, it then follows that $k=1/2$.
Therefore, if there is such a $g(z)$, then $f(z)-g(z)$ is a weight 1/2 harmonic weak Maass form which has a nonvanishing nonholomorphic part, which also has the property
 that $f^{+}(z)-g(z)$ has no exponential
singularities at any roots of unity. This contradicts
Theorem~\ref{mainthm3}.

\section{Proof of Theorem \ref{suppthm3}}\label{ProofSuppThm3}
Ramanujan found, and Watson proved, a number of relations between mock theta functions \cite{Watson}. In particular, Watson showed that for the mock theta function 
\begin{equation}\nu(q):=\sum_{n\geq0}\frac{q^{n(n+1)}}{(-q;q^2)_{n+1}}\end{equation} we have the relation 

\begin{equation}\label{Ram-Wat}\nu(\pm q)\pm q\omega(q^2)=\frac12\cdot\left(\displaystyle\sum_{n\in\Z}q^{n(n+1)} \right)\cdot\prod_{n\geq0}(1+q^{2n}).\end{equation}
The idea is then that $\nu(q)$ may be expressed in a different form due to a result of Fine. Namely, the following is a limiting case of the famous Rogers-Fine identity (see (7.31) of \cite{Fine}):
\begin{equation}\label{R-F}
F(b/t,0;t)=\frac{1}{1-t}\sum_{n\geq0}\frac{(-b)^nq^{\frac{n^2+n}{2}}}{(tq;q)_n},\end{equation}
where $F(a,b;t):=\sum_{n\geq0}t^n\frac{(aq;q)_n}{(bq;q)_n}.$
Taking $b=-1$, $t=-q^{\frac12}$ and letting $q\rightarrow q^2$, we find by (\ref{R-F}):
\begin{equation}
\nu(q)=\sum_{n\geq0}(-q)^n(q;q^2)_n
.
\end{equation}
It is thus apparent that at odd order roots of unity, the sum on the right-hand side is finite and $\nu(q)$ converges, whereas for a primitive order $k$ root of unity with $k\equiv2\pmod4$, it is clear that $\nu(-q)$ becomes a finite sum. Choosing the sign appropriately in (\ref{Ram-Wat}) gives Theorem \ref{suppthm3}.

