关于堆垒素数几个典型定量问题研究

本项目主要围绕以下几类典型堆垒素数定量问题展开研究：（1）结合近几年Bourgain、Wooley等人在Vinogradov均值估计上的重大进展，研究一些重要的素变数指数和及其积分均值的定量估计。（2）结合近期算术组合以及Dirichlet L-函数零点分布的有关进展，考虑一些带系数的素变数方程的可解性问题，进而考虑其最小素数解的定量上界估计。（3）结合近期在二次、三次型上的有关研究及进展，研究一些重要的二次、三次不定型的最小素数零点的上界估计，并继续考虑一些含有交叉项的二次、三次素变数方程的最小素数解的上界估计。（4）借鉴Goldston-Pintz-Yildirim、Maynard和Tao等人在筛法上的研究思想和方法，利用项目组成员多年的工作积累，考虑一些经典素数分布的定量结果和改进应用。比如，算术数列以及短区间上的素数定理余项的定量估计问题。

The purpose of this project is to study the following several typical quantitative problems in the field of additive prime number theory. (1) Using the latest improvements of Bourgain, Wooley et al. in the estimates of the Vinogradov's mean values, we will do some researches of quantitative estimates for some important prime variables exponential sums and integral mean values. (2) Combining the arithmetic combination and the latest progress of zeros of Dirichlet L-function, consider the solvability for some prime variables equation with coefficients, furthermore , we will study the estimates of quantitative upper bounds for the least prime solutions. (3) Combining the latest researches and improvements for quadratic and cubic forms, research the quantitative upper bounds of the least prime zeros for some important quadratic and cubic indefinite forms, furthermore, consider upper bound estimations of the least prime solutions of some quadratic and cubic prime variables equations. (4) Making use of the idea and sieve method of Goldston-Pintz-Yildirim, Maynard and Tao, by the accumulation of the team member, we will consider some quantitative results and applications of classical distribution of primes, such as, quantitative estimates for the error terms of prime theorem in arithmetic progression and short intervals.