两类指数和的相关性质及应用

The study on the properties of two kinds of exponential sums such as Kloosterman sums and the two-term exponential sums occupies a pivotal position in additive number theory, analytic number theory, modular forms, elliptic curves, and coding theory. And many famous number theoretic problems such as "Linnik conjecture", "Selberg's eigenvalue conjecture", "Sato-Tate conjecture" and "Waring's problem" are closely related. In the project the upper bound, high power mean and hybrid mean of the general Kloosterman sums and the two-term exponential sums with character are to be studied by using the methods of estimating the exponential sums and character sums. Many techniques are to be applied in order to obtain some sharp asymptotic formulae, through which the distributive properties of the two sums are revealed, and the theory involving them is complemented. As an application, the coding properties of Kloosterman code are to be explored and promoted. Furthermore, some new linear codes based on the high power Kloosterman and the two-term exponential sums are to be constructed, with an attempt to provide the coding theory new research contents and methods of development.

指数和研究中关于Kloosterman和及二项指数和性质的讨论在加法数论、解析数论、模形式、椭圆曲线以及编码研究中占有举足轻重的位置，并和很多数论难题如"Linnik猜想"、"Selberg特征值猜想"、 "Sato-Tate猜想"以及"华林问题"等密切相关。本项目主要针对带特征Kloosterman和及二项指数和的上界估计、高次均值、混合均值等性质进行深入研究，拟采用指数和及特征和估计方法，始终把握在指数和及特征和估计中"均值估计优于单个估计"的原则，并结合"分段"、"分类"、"凑项"、"整合"、"转换"等技巧，以期获得一些较强渐近公式，进一步揭示这两类指数和的均值分布规律，补充和拓展指数和的相关理论。作为应用，拟对Kloosterman 码的编码性质进行探索和推广，并基于高次Kloosterman 和及二项指数和尝试构造新的线性码，以期为编码理论的研究和发展提供新的研究内容与方法支持。

关于Kloosterman 和及二项指数和性质的讨论在解析数论研究中十分重要。本项目主要研究了广义二项指数和的高次均值、经典Kloosterman和的双线性型、广义r次Kloosterman和的混合均值、广义二次Gauss和的混合均值、超级Cochrane和的上界估计、一些特殊短区间及一般短区间上的高维D. H. Lehmer问题等。研究过程中采用了指数和及特征和估计方法，获得了一些较强的渐近公式或精确的计算公式，进一步揭示了这两类指数和均值分布规律，补充和拓展了指数和的相关理论。