2025/11/01
Heuristically, this is about simple understandings of real-variable Kloosterman sums from my research [6]. This article is not intended to give rigorous proofs but just some directions.
We still use the notation $q=e(z)=e^{2\pi i z}$. When defining Poincaré series or Maass-Poincaré series, we use the coset $\Gamma_\infty \setminus \Gamma$. Usually it doesn't matter for the choice of coset representation, but now since our seed function $e(r^2 z/2)$, $\varphi_{s,k}(r^2 z/2)$ are not periodic for $z\rightarrow z+2$, we need to restrict the coset representation to be: \[\gamma\in \Gamma \text{ such that } \gamma \mathcal{F} \subseteq [-1,1]\times i[0,\infty).\]
This is the idea from (Martin Stoller. Fourier interpolation from spheres, Trans. Amer. Math. Soc. 374, 2021). Currently we have the easy case: $\Gamma=\Gamma(2)$, and the coset representation directly corresponding to the restriction $\gamma=\begin{pmatrix} a&b\\c&d \end{pmatrix}$ with $|a|<|c|$, $|b|<|d|$. We may follow this kind of definition for the other congruence subgroups, while they may not be that clear. For example, for $\Gamma(1)$ we need some extra work.
Anyway, in $\Gamma(2)$ we can define the real-variable Kloosterman sums in a clear way. We always focus on $c>0$. We use $S$ to indicate that this is the (classical) Kloosterman sum, and use $K$ to indication that this allows real variable. \[\text{if }2|c,\quad K(r,n,c,\nu_{\Theta}^{2k} ) =\sum_{\substack{-c < a < c \\ d\pmod {2c}^*\\ad\equiv 1(2c) }} \varepsilon_d^{2k} (\tfrac{2c}d)^{2k} e\lp \frac{ra+nd}{2c}\rp, \] \[\text{if }2\nmid \tilde{c},\quad K(r,n,\tilde{c},\nu_{\Theta}^{2k} ) =\sum_{\substack{-b < d < b \\ 2|a \pmod{2\tilde{c}},\, 2|d\\ ad\equiv 1(\tilde{c})}} e(\tfrac k4) \varepsilon_{\tilde{c}}^{-2k} (\tfrac{2d}{\tilde{c}})^{2k} e\lp \frac{ra+nd}{2\tilde{c}}\rp. \]
Here are some observations we made, where we only write the series for $K(r^2,n,c,\nu_\Theta^{2k})$ but the conclusion is similar for $K(r^2,n,\tilde{c},\nu_\Theta^{2k})$.
(1) For weight $k=2$.
(1.1) When $n\neq 0$, the following series converge for $s=1$ and have good tail estimates: \[\sum_{2|c>0}\frac{K(r^2,n,c,\nu_\Theta^4)}{c}J_{2s-1}\left(\frac{2\pi|r^2n|^{\frac 12}}{c}\right),\quad \sum_{2|c>0}\frac{K(r^2,n,c,\nu_\Theta^4)}{c}I_{2s-1}\left(\frac{2\pi|r^2n|^{\frac 12}}{c}\right). \]
(1.2) When $n=0$, the following series has a pole when $s\to 1$ with residue \[\operatorname*{Res}_{s=1} \sum_{2|c>0}\frac{K(r^2,0,c,\nu_\Theta^4)}{c^{2s}}=\frac{2\sin(\pi r^2)}{\pi^3r^2}. \]
(2) For weight $k=3/2$.
(2.1) When $n\neq -m^2$ for any $m\in \mathbb Z$, the following series converge for $s\in[3/4,1]$ and have good tail estimates: \[\sum_{2|c>0}\frac{K(r^2,n,c,\nu_\Theta^3)}{c}J_{2s}\left(\frac{2\pi|r^2n|^{\frac 12}}{c}\right) \]
(2.2) When $n=0$, the following series has a pole when $s\to 3/4$ with residue \[\operatorname*{Res}_{s=3/4} \sum_{2|c>0}\frac{K(r^2,0,c,\nu_\Theta^3)}{c^{2s}}=e(-\tfrac 18)\frac{\sin(\pi r^2)}{\pi^2r \sinh(\pi r)}. \]
(2.3) When $n=-m^2$ for some $m\in \mathbb Z$, $m\geq 1$, the following series has a pole when $s\to 3/4$ with residue \[\operatorname*{Res}_{s=3/4} \sum_{2|c>0}\frac{K(r^2,-m^2,c,\nu_\Theta^3)}{c}I_{2s-1}\left(\frac{2\pi|r^2n|^{\frac 12}}{c}\right)=e(-\tfrac 18)\frac{4|rm|^{\frac 12} \sin(\pi r^2)}{\pi^2r \sinh(\pi r)}. \]
Let $\Theta(z)=\sum_{n\in \mathbb Z} e^{\pi i n^2 z}=\sum_{m=0}^\infty r_1(m) e^{\pi i m z}$. The reason for the residues in part (2) is because \[\sum_{m=0}^\infty \frac{(-1)^m \,r_1(m)}{r^2+m}=\frac 1{r^2}+\sum_{m=1}^\infty \frac{2(-1)^m}{r^2+m^2}=\frac{\pi }{r\sinh(\pi r)}\quad (r>0). \]
We end with three questions:
(Q1) Clearly, the residues and convergence are related to the coefficients of $\Theta^{4-2k}$ for $k=2,3/2$. Why?
(Q2) How do we deal with the case $k=0,1/2$ where $s\to 0,1/4$? The analytic continuation of these real-variable Kloosterman sums will be interesting. Moreover, for the central point, how do we deal with $k=1$ and $s\to 1/2$?
(Q3) Why $\Gamma_\Theta$ and $\Gamma(2)$ are special here? What's the reasonable generalization for other congruence subgroups, and what will be the requirement for the sum on $a$ then?
Please contact me if you are interested and have any ideas.