关于指数和、特征和的相消性及相关问题研究在解析数论、加法数论、模形式以及椭圆曲线研究中十分重要，并和很多难题如Linnik猜想、Sato-Tate猜想、华林问题、算术序列中的除数问题等密切相关。本项目主要研究Kloosterman和、二项指数和、特征和的加权均值及高次均值，以及这些和式在一些特殊区间或特殊数集的上界估计等，以揭示指数和、特征和的相消性规律。作为指数和、特征和相消性的重要应用，拟将所得研究成果应用于一些经典数论问题如算术序列中的除数问题、无平方因子数问题、完全平方数问题、整数及其逆的分布问题、D.H. Lehmer问题等。本项目力争获得一些改进甚至突破性结果，丰富和完善相关理论。

Cancellations among exponential sums, character sums, and relevant problems occupy a pivotal position in analytic number theory, additive number theory, modular forms, and elliptic curves. And many famous number theoretic problems such as Linnik conjecture, Sato-Tate conjecture, Waring's problem, divisor problem in arithmetic progressions, are closely related to these famous sums. This project mainly studies weighted mean and high-power mean of Kloosterman sums, exponential sums with binomials, and character sums. The project also studies the upper bounds of these sums on some special intervals or special number sets, through which possible cancellations are revealed. As important applications of cancellations among exponential sums or character sums, the project intends to apply the research results to some classical number theoretic problems such as the divisor problem in arithmetic progressions, the square-free numbers problem in arithmetic progressions, the square-full numbers problem in arithmetic progressions, the distribution of an integer and its inverse, and D.H. Lehmer problems, and so on. The project strives to obtain some improvements and even derive some breakthrough results, attempting to supplement and enrich relevant theories.