素数分布理论是数论研究的核心内容。素数是整数乘法结构中的砖瓦，而华林-哥德巴赫问题的本质是揭示整数与素数之间的加性联系、探索素数分布的深层次规律。近年来，有关素数分布的研究取得了一系列突破。现代数论研究的先进工具，如：渐近筛法理论、自守形式理论、遍历论等在这些突破中发挥着重要的作用。申请人在扩大圆法的主区间、小区间上的华林-哥德巴赫问题等方向上取得了一些成果。基于这些工作的成功和我们已有的工作基础，我们设计了如下新的技术路线来研究华林-哥德巴赫问题：即一方面寻求Dirichlet多项式均值的新估计继而进一步扩大圆法的主区间，另一方面改进筛法中两类双线性指数和的估计，从而解决一些重要的华林-哥德巴赫问题。证明这一技术路线是可行的，是本项目拟解决的关键问题。

The theory of distribution of primes is the core of number theory. Prime is the cornerstone in the multiplicative structure of integers, while the essence of Waring-Goldbach problem is to show the additive relation between primes and integers, and to explore the advanced law of distribution of primes. Recently there are a lot of breakthroughs in the study of distribution of primes. Advanced tools in modern number theory, such as the theory of asymptotic sieves, automorphic forms and ergodic theory, play very important roles in these breakthroughs. In the meantime, the applicant has made some work on the enlarged major arcs of circle method and on the Waring-Goldbach problem in short intervals. Based on the success of these works and our study experience, we design the following technical route to study the Waring-Goldbach problem. On one hand, we investigate new mean-value estimates for Dirichlet polynomials to get still larger major arcs. On the other hand, we need to improve the upper bound of two kinds of bilinear sums arising from sieve methods. To prove our technical route is feasible is the crucial problem that this program needs to solve.