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For $m,n,c\in\mathbb{N}$ let $K(m,n;c)$ S(m,n;c)$be the Kloosterman sum $$K(m,n;c)=\sum_{a=1, S(m,n;c)=\sum_{a=1, \gcd (a,c)=1}^ce\left(\frac{ma+n\overline{a}}{c}\right).$$ 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? 2 Added a few details; deleted 1 characters in body For$m,n,c\in\mathbb{N}$let$K(m,n;c)$be the Kloosterman sum $$K(m,n;c)=\sum_{a=1, \gcd (a,c)=1}^ce\left(\frac{ma+n\overline{a}}{c}\right).$$ 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. Does anyone know if there are I am working on an application for which I need an analogous results bound for the sum $$\sum_{p\le x}\frac{S(m,n;p)}{p},$$ where$c$p$ runs over prime numbersup . Does anybody know of a way to some obtain a good bound (i.e. better than the Weil bound) for this sum?
For $m,n,c\in\mathbb{N}$ let $K(m,n;c)$ be the Kloosterman sum $$K(m,n;c)=\sum_{a=1, \gcd (a,c)=1}^ce\left(\frac{ma+n\overline{a}}{c}\right).$$ 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. Does anyone know if there are analogous results where $c$ runs over prime numbers up to some bound?