Question

For $m,n,c\in\mathbb{N}$ let $S(m,n;c)$ be the Kloosterman sum $$S(m,n;c)=\sum_{a=1, \gcd (a,c)=1}^ce\left(\frac{ma+n\overline{a}}{c}\right).$$ The Kuznetsov Trace Formula allows us to obtain bounds for sums of the form $$\sum_{c\le x,~ c=0\mathrm{mod} q}\frac{S(m,n;c)}{c},$$ which are better than those obtained by simply applying Weil's inequality. I am working on an application for which I need an analogous bound for the sum $$\sum_{p\le x}\frac{S(m,n;p)}{p},$$ where $p$ runs over prime numbers. Does anybody know of a way to obtain a good bound (i.e. better than the Weil bound) for this sum?

Answer 1

No, this is a well-known open problem. One doesn't even know that the sign of $S(1,1;p)$ changes infinitely often... The best that has been achieved are estimates restricted to moudli $c$ with a bounded number of prime factors, by combining sieve methods with automorphic forms (and some average forms of Sato-Tate), see the papers of Fouvry and Michel on the topic.