Mean square estimate for the Kloosterman sums

Question

For $m,n\in \mathbb{N}$, denote the Kloosterman sum $$S(m,n;c)=\sum_{a\bmod c}e\left( \frac{ma+n \overline{a}}{c}\right),$$where $\overline{a}$ denotes the multiplicative inverse of $a\bmod c$.

Does any expert here know something upon the non-trivial power-saving bound for the mean square estimate of the Kloosterman sum, that is, for $m,n$ fixed, whether or not one has an estimate $$\sum_{c\le x}\frac{|S(m,n;c)|^2}{c}\ll_{m,n}\,\,x^{\theta}\tag{$\ast$}$$ for some $\theta<1$?

As far as we know, the non-trivial bound for the first moment is due to Kuznetzov who showed $$\sum_{c\le x}\frac{S(m,n;c)}{c}\ll_{m,n}\,\,x^{\frac{1}{6}}\,\log ^{\frac{1}{3}}x.$$ The Linnik’s conjecture asserts that $$\sum_{c\le x}\frac{S(m,n;c)}{c}\ll_{m,n}\,\,x^{\varepsilon}$$ for any $\varepsilon>0$. For detailed description on the first moment estimate, one may see Sarnak and Tsimerman's paper: https://www.researchgate.net/publication/225888754_On_Linnik_and_Selberg%27s_Conjecture_About_Sums_of_Kloosterman_Sums

Recently I encounter this tricky sum as in ( $\ast$), for which one need to get a saving, compared with the trivial bound $x^{1+\varepsilon}$. There does not seem to be a reference available in the literature. If any expert here have some strategies or references, please give a guide. Many thanks.

Your any opinions are highly appreciated. Thanks in advance.

Answer 1

You cannot have estimates like (*) for any $\theta<1$. Fouvry and Michel showed that (see Theorem 1.2 there) $$ \sum_{c\le x} |S(m,n;c)|/\sqrt{c} \gg_k \frac{x}{\log x} (\log \log x)^k, $$ for any natural number $k$. By Cauchy-Schwarz one also gets a lower bound for the second moment.

They also note that one can get an upper bound for this quantity, saving a power of log. I would guess that $$ \sum_{c\le x} |S(m,n;c)|^2/c $$ is of the order $x$ (assume $mn \neq 0$).