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<p}\Big|\frac{1}{\sqrt{p}}\sum_{0\leq n < H}t(n)\Big|. \] In this very general context one of the most important results is the P\'olya-Vinogradov bound \[ M(t)\leq \left\|K\right\|_{\infty}\log 3p. \] where $K:\mathbb{F}_{p}\rightarrow\mathbb{C}$ is the normalized Fourier transform of $t$. In this paper we provide a lower bound for incomplete Kloosterman sum, namely we prove that for any $\varepsilon \>0$ 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 ${M(e(\tfrac{ax+\overline{x}}{p}))}{a\in\mathbb{F}_{p}^{\times}}$.