基于素数差集与Riemann zeta函数的性质研究

The set of differences among primes and the Riemann zeta-function are very important in the research of number theory, and have a very close relationship. Let D be the set of positive even numbers that can be expressed in infinitely many ways as the difference of two primes. This project aims at the study on how large the set D, the most likely difference among primes, the proportion of zeros on the critical line and moments of the Riemann zeta-function. It is proved recently that the least number in D is not more than 246. The problem on how large the set D is very important when one considers the Kronecker’s conjecture, however it has not been proved that D is a subset with a positive density in the set of all even number. By combining the GPY sieve method and earlier obtained results, this project study the set D and the distribution of most likely differences among primes deeply. In addition, based on plentiful works on moments of the Riemann zeta-function obtained recently by people, and also on the proposer’s preceding work on estimation on error terms, this project also focuses on enlarging the proportion of zeros on the critical line and making a breakthrough on moments of the Riemann zeta-function by employing the mollifier with a more general form and a longer length. By deeply studying on these problem together, this project also focuses on making a breakthrough at some new ideas and creative technique to achieve the goal.

素数差集和Riemann zeta函数是数论中非常重要的研究对象，而且关系紧密。记D为能表示成无穷对素数差的正偶数构成的集合。本项目将研究D的大小、频率最高的素数差值分布以及Riemann zeta函数在临界线上零点比例和其积分均值问题。最近人们在D的下界上取得重大突破，证明了其下界不超过246。而从Kronecker’s猜想出发，D的大小尤为重要，但至今仍无法证明D是偶数集的正密度子集。本项目从GPY筛法出发，结合之前取得的工作，力争进一步提高D的范围和挖掘频率最高的素数差值信息。本项目还将从控制余项出发，通过赋予mollifier更一般的形式和更长的长度，争取进一步提高Riemann zeta函数临界线上零点比例，并力争在积分均值问题上取得突破。这一路径基于申请人在余项控制上已取得一些进展和国际上最近关于积分均值的丰富成果。本项目综合研究以上问题，力争在理论和方法上取得突破，实现目标。

素数分布与L函数是数论中非常重要的研究对象，且两者关系紧密。在素数分布和L函数方面的研究进展将对很多数论问题起到推动作用。在本课题的资助下，我们研究了素数差值频率问题、Waring-Goldbach问题以及编码应用等素数分布相关问题；研究了包括L函数均值、零点分布等L函数性质，在Math. Ann., Proc. Lond. Math. Soc等著名学术期刊上发表SCI论文11篇。主要取得了如下研究成果：基本解决了Dirichlet L函数中心点处四次均值渐近公式猜想中的渐近公式问题；在Dirichlet L函数层面上，得到了满足Riemann猜想零点的新比例，消除了Dirichlet L函数与Riemann zeta函数上的研究差距；在Dirichlet L函数与Dedekind zeta函数单零点相关问题的研究上推进了已有的结果；给出了一类扭模L函数的实数次均值的精确界；在三次Waring-Goldbach问题上取得了重要进展；得到了k元素数组频率最高的差值集合的一些基础性分布性质；作为应用在循环码、量子码的构造上取得新进展。这些研究工作加深了对素数分布与L函数的理论认识，并对相关理论的发展起到了积极推动作用。