On the size of the maximum of incomplete Kloosterman sums

Abstract

Let $t:\mathbb{F}_{p}\rightarrow\mathbb{C}$ be a complex valued function on $\mathbb{F}_{p}$. A classical problem in analytic number theory is to bound the maximum of the absolute value of the incomplete sum \[ M(t):=\max_{0\leq H 0$ there exists some $a\in\mathbb{F}_{p}^{\times}$ such that \[ M(e(\tfrac{ax+\overline{x}}{p}))\geq \Big(\frac{1-\varepsilon}{\sqrt{2}\pi}+o(1)\Big)\log\log p. \] Moreover we also provide some result on the growth of the moments of $\{M(e(\tfrac{ax+\overline{x}}{p}))\}_{a\in\mathbb{F}_{p}^{\times}}$.