素数分布是数论的基础研究领域之一。小区间上的素数定理反映了素数的局部性质，在很多堆垒数论问题中成为一个合理而常见的限制。其内在联系在于，两者的改进均依赖零点密度估计。其中，几乎相等素变量的华林-哥德巴赫问题是深化华罗庚定理的一个重要研究方向，这一问题直接结合了小区间上的素数分布与素数堆垒问题。本项目旨在利用圆法、筛法等工具，深入理解两者的关系。以几乎相等素变量的华林-哥德巴赫问题为核心，探索其指数在[0,1)上的容许性和半容许性，以得到优于现有基于广义黎曼猜想（GRH）的结果。其筛法改进的经验可以应用于小变量的华林-哥德巴赫问题、Carmichael素数等素数堆垒问题的研究。最后，通过推广Vinogradov均值定理，改进齐次丢番图方程的有理解逼近，构造优于弱逼近定理的逼近性质。

The study of distribution of primes is a fundamental theme in number theory. The prime number theory in short intervals reveals the local properties of prime numbers, which leads to a reasonable and common limitation in many additive problems. The intrinsic link is that the improvements of both problems depend on the zero density estimation. The Waring-Goldbach problem with almost equal primes is an important refinement of Hua's theorem, which directly combines the distribution of primes in short intervals and their additive theory. The purpose of this project is to understand the their relationship by tools including the circle method and the sieve method. The project will mainly focus on the Waring-Goldbach problems with almost equal primes. By studying the admissibility and semi-admissibility of the exponent in [0,1), we aim to establish theory beyond generalized Riemann hypothesis(GRH). Similar sieve method applies to the Waring-Goldbach problems with small variables, Carmichael numbers, etc.. Finally, through the generalization of Vinogradov's mean value theorem, we investigate the rational approximation of homogeneous Diophantine equation, to approach properties beyond weak approximation.