On the size of the maximum of incomplete Kloosterman sums

Abstract

Let $t:{\mathbb{F}}_p\to \mathbb{C}$ 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 Pólya-Vinogradov bound \[ M(t)\leq \left\|K\right\|_{\infty}\log 3p. \] where $K:{\mathbb{F}}_p\to \mathbb{C}$ 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 $\epsilon >0$ there exists some $a\in {\mathbb{F}}_p^{\times }$ such that \[ M(e(\tfrac{ax+\overline{x}}{p}))\geq \Big(\frac{1-\varepsilon}{\sqrt{2}π}+o(1)\Big)\log\log p. \] Moreover we also provide some result on the growth of the moments of $\{M(e(\frac{ax+\overline{x}}{p})){\}}_{a\in {\mathbb{F}}_p^{\times }}$.

References in corpus (2)

• Gaussian distribution of short sums of trace functions over finite fields
• On the distribution of the maximum of cubic exponential sums