Twisted higher moments of Kloosterman sums

Abstract:

Let $\chi$ be a nontrivial Dirichlet character modulo an odd prime $p$. Write \begin{equation*} S(a)=\sum \limits _{x=1}^{p-1}e(\frac {x+ax^{-1}}{p})=2\sqrt {p}\cos \theta (a). \end{equation*} We shall prove \begin{equation*}\sum \limits _{a=1}^{p-1}\chi (a)S(a)^{2}=\chi (-1)g(\chi )^{2}J(\chi ,\bar {\chi }^{2}) \end{equation*} and, for complex $\chi$, \begin{equation*}|\sum \limits _{a=1}^{p-1}\chi (a)\frac {\sin (k+1)\theta (a)}{\sin \theta (a)}|\leq c(k)\sqrt {p},\ k>0, \end{equation*} where $c(k)$ is a constant depending only on $k$.