# Mean square estimate for the Kloosterman sums

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.

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 $$mn \neq 0$$).