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I asked this question on Mathematics stack exchange but didn't get a response, so I ask here too. Let $\chi$ be the non-trivial quadratic character of $\mathbb{F}_q$, and let $f(x)$ be a square-free ...

There is a basic question regrading the 3-dimensional hyper-Kloosterman sum which needs some help from the experts here: For any integers $q,h \in \mathbb{N}$, how to estimate the sum: $$\sideset{_{}^{...

There is a basic question regrading the mixed 3-dimensional hyper-Kloosterman sum: For any positive integer $n$ not divisible by $p$, how to prove $$\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y ,z\bmod p} \...

I encounter a tricky sum like the Kloosterman sum $${\sum_{x \mod P}}^\ast e\left(\frac{ax+\overline{x}}{P}+\frac{lx^2}{P^2}\right),$$ where $l$ is a positive integer co-prime with $P$ and here $P$ ...

Denote the classical Kloosterman and Salié sums, respectively, as $KL(a,b) = \sum_{r \in F_*} e(ar+\frac{b}{r})$ and $SL(a,b) =\sum_{r \in F_*} \chi(r) e(ar+\frac{b}{r})$, where $\chi(\cdot)$ is the ...