I've been studying the book "Topics in Classical Automorphic forms" by Henryk Iwaniec and I am stuck at a bound. For even $k$ and any $3/4 < \sigma < 1$ the book claims the following for the Kloosterman sum $S(n,n,c)$ where $\epsilon>0 $ is fixed, as $n$ gets large. (page 73)
$$\sum_{c>0} c^{-2\sigma}|S(n,n,c)| \ll n^{\epsilon} $$
I've been trying to prove this using the Weyl bound $|S(m,n,c)| \le \gcd(m,n,c)^{1/2} c^{1/2}\tau(c)$ but I haven't managed to do so. Maybe I am missing something basic but in order to create a convergent series the bounds for the $\tau$ function seems to exceed any hopes I have for that.
Since $|S(n,n;c)| \leq (n,c)^{1/2} c^{1/2} \tau(c)$ and $\tau(c) \ll_{\varepsilon} c^{\varepsilon}$, this sum is $$\ll_{\varepsilon} \sum_{c = 1}^{\infty} \frac{(n,c)^{1/2}}{c^{2\sigma - 1/2 - \varepsilon}}.$$ Now write this sum over $c \in \mathbb{N}$ as a double sum over $m \mid n$ and over $c \in \mathbb{N}$ for which $(c,n) = m$ (that is, break up the sum over $c$ based on the $\gcd$ of $c$ and $n$). Having done this, make the change of variables $c \mapsto mc$, since $(c,n) = m$ implies that $m \mid c$. We get $$\sum_{m \mid n} \frac{1}{m^{2\sigma - 1 - \varepsilon}} \sum_{\substack{c = 1 \\ (c,\frac{n}{m}) = 1}}^{\infty} \frac{1}{c^{2\sigma - 1/2 - \varepsilon}}.$$ The inner sum over $c$ converges absolutely since $\sigma > 3/4$. The outer sum is $\ll_{\varepsilon} n^{\epsilon}$.