Goldbach-Linnik问题相关方程组的研究

According to the even Goldbach problem in the analytic number theory, this program is to research systems of equations related to the Goldbach-Linnik problem. In order to show that parts of them are solvable, calculate the exact numerical result in the above solvable systems of equations including the number of powers of 2 and the up bound of small prime solutions, we apply the two-dimensional circle method, the large sieve inequality, estimate on the zeros of Dirichlet L-functions and so on. Through this research, we expect to show the feature that these systems of equations related to the Goldbach-Linnik problem with some necessary conditions on the coefficients are solvable, and establish the whole solution system of those systems of equations based on their structure characteristic. The research achievement will be meaningful in the research of solutions system of systems of linear equations with several variables and integral coefficients, and the distribution of prime solutions in the above systems of equations. Moreover, the research will provide the approach in the research of some special prime variable linear equations for example the even Goldbach problem.

针对解析数论中的偶数哥德巴赫问题，本项目以Goldbach-Linnik问题相关方程组为研究对象，采用二维圆法、大筛法型不等式、Dirichlet L 函数的零点密度估计等方法，研究该方程组解的存在性，确定解的结构，并深化该问题，确定2的幂次的个数以及小素数解的上界。项目预期将揭示Goldbach-Linnik问题相关方程组只需系数满足部分必要条件就存在解的特质，并根据该类方程组解的结构特点建立其全面的解体系。研究成果对阐明整系数多变量线性方程组求解理论，揭示该类方程组素数解分布规律具有重要意义，同时为研究特殊的整系数素变量线性方程，比如偶数哥德巴赫问题，提供思路。

本项目在Linnik方法趋近Goldbach问题的基础上，研究Golbach-Linnik相关方程组的解的存在性、解组中2的幂次的个数以及解组中小素数解的上界问题。通过利用二维圆法、区间上的指数和估计等方法，揭示出系数只需满足一系列必要条件的四个素变量的Goldbach-Linnik方程组都是有解的这一事实，计算出一类系数给定的该方程组的2的幂次的个数，还给出了此类方程组的小素数解一般上界。其中的关键数据有:计算上述2的幂次个数269所需的主区间上积分估计下界的系数是0.617023，次主区间上积分估计上界的系数是405.814，第三区间上关键常数lambda是0.975805; 推算出带有系数a(ij)且系数满足组成行列式非零的相关方程组的2的幂次是只与系数相关的函数。研究成果对阐明正整系数多变量线性方程组求解理论，揭示该类方程组素数解分布规律具有重要意义；同时为研究特殊的整系数素变量方程，比如Goldbach-Linnik问题，提供思路。