Incomplete Kloosterman sums and Hooley's R$^*$-Conjecture

Incomplete Kloosterman sums and multiplicative inverses in short intervals

We investigate the solubility of the congruence xy=1 (mod p), where p is a prime and x,y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums.

On the size of the maximum of incomplete Kloosterman sums

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 Pólya-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…

Short Kloosterman sums to powerful modulus

We obtain the estimate of incomplete Kloosterman sum to powerful modulus $q$. The length $N$ of the sum lies in the interval $e^{c(\log{q})^{2/3}}\le N\le \sqrt{q}$.

Sumsets of reciprocals in prime fields and multilinear Kloosterman sums

We obtain new results on additive properties of the set $$ I^{-1}= \{x^{-1}: \quad x\in I\} $$ where $I$ is an arbitrary interval in the field of residue classes modulo a large prime $p$. We combine our results with multilinear exponential sum estimates and obtain new results on incomplete multilinear Kloosterman sums.

函数域上迹函数的短和及其应用

Short sums of trace functions over function fields and their applications

研究函数域上迹函数的短和及其应用，解决了多项式无法表示的最小剩余类问题和Kloosterman和的方差问题。

短区间内D. H. 莱默的一个问题. I

A problem of D. H. Lehmer in short intervals. I

研究与模q乘法逆元奇偶性不同的整数个数问题，利用算术指数对和素变量不完全克洛斯特曼和估计，在特定条件下得到短区间内此类整数个数渐近公式并证明素变量版本，改进了前人结果。