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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 ...

Let $q$ be a power of prime $p$ and $\zeta_p$ be the complex $p$ th root of unity. We denote by $\mathbb{F}_q$ the finite field of $q$ elements and by $Tr$ the absolute trace function $\mathbb{F}_q\...