On the size of the maximum of incomplete Kloosterman sums

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 $$M(t):=\underset{0\le H<p}{\max }\mid \frac{1}{\sqrt{p}}\sum _{0\le n<H}t(n)\mid \mathrm{.}$$ In this very general context one of the most important results is the P'olya-Vinogradov bound $$M(t)\le {\parallel K\parallel }_{\mathrm{\infty }}\log 3p\mathrm{.}$$ 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 $$M(e(\frac{ax+\bar{x}}{p}))\ge (\frac{1-\epsilon }{\sqrt{2}\pi }+o(1))\log \log p\mathrm{.}$$ Moreover we also provide some result on the growth of the moments of $\{M(e(\frac{ax+\bar{x}}{p})){\}}_{a\in {\mathbb{F}}_p^{\times }}$.