素变数指数和估计在素变量堆垒问题的研究中起着重要作用. 张益唐在孪生素数猜想上的突破和 Sarnak 猜想的提出为素数相关问题的研究带来了新的生机. 在几乎相等的华林-哥德巴赫问题和小区间上的三元二次型表素数等问题的研究中, 申请人曾取得了一定的成果. 在此基础上, 本项目将主要考虑小区间上的素变数指数和估计, 并结合以下两方面问题进行研究: 1. 几乎相等的华林-哥德巴赫问题; 2. 与 Bombieri 型定理相关的问题. 我们拟采用的研究方法是: 改进多重非线性指数和估计, 进而优化小区间上的素变数指数和上界; 通过引入筛函数改进 Dirichlet 多项式的均值估计.

Estimates of exponential sums over primes plays an important role in research of additive problems with primes in analytic number theory. The breakthrough of Zhang Yitang and Sarnak’s conjecture bring in some new ideas and methods. The applicant has some research on Waring-Goldbach problems and the distribution of primes in the ternary quadratic form in short intervals. In this project, we will consider exponential sums over primes in short intervals combined with the following problems: 1. Waring-Goldbach problems with almost equal primes; 2. Some problems related to Bombieri-type theorem. We will apply the following techniques. For one thing, we investigate a mean-value estimates of Dirichlet polynomials in combination with sieve method. For the other thing, we will refine estimates of the nolinear sums appearing in the investigation of exponential sums over primes.