The Higher Rank Selberg Sieve and Applications

1947年，阿特尔·塞尔伯格在筛理论中发现了一种新方法，使这一学科发生了革命性的变化。这种方法现在被称为Selberg筛，并已被用于从经典的孪生素数问题到更复杂的代数变异点数问题的一系列应用中。最近，Maynard和Tao（独立地）应用了Selberg筛的“高阶”版本的一个特例来改进和简化Zhang关于连续素数之间无限多有界间隙的开创性工作。在与我的博士生Akshaa Vatwani的合作中，我开发了一种通用的更高级别的经典塞尔伯格筛。梅纳德和陶的工作现在看来是这种更普遍筛子的一个特例。显然，这种新的筛选方法将有进一步的应用，我们计划在未来几年将其应用于各种各样的问题。看起来似乎会有一些潜在的应用程序来解决Artin原始根问题。此外，还可以为这个高阶筛制定一个数字字段版本。我的学生已经在她正在进行的博士论文中研究了这种可能性，并获得了一些关于高斯素数之间有界间隙的有趣结果。也就是说，有一个固定的数B使得有无限多个高斯素数a+bi和c+di使得|a-c|和| B -d|都以B为界我们还得到了一个新的陶伯利定理的证明我们期望在这个情况下得到更多的结果。很明显，这些最近的发现代表了筛理论的一个转折点。事实上，高阶Selberg筛与经典Selberg筛之间的关系类似于多变量微积分和单变量微积分的发现。例如，在经典的Selberg筛中，为了证明存在无穷多个孪生素数，将筛法应用于单序列n（n+2）而不是二元序列（n, n+2）。历史上有趣的是，早在1969年，塞尔伯格就在他的一篇论文的末尾提出了“更高级别”的筛选方法，但很明显，这个想法一直没有被注意到，直到最近被梅纳德和道重新提起。虽然单变量理论可以研究多变量函数，但多变量理论的结构更丰富。经典筛和高级筛的情况也类似。***我们还期望高阶Selberg筛在研究满足Chebotarev条件的素数间隙，特别是当基域不是有理数域时的其他应用。我的博士生，王鹏杰，现在正在研究Artin l系列，我们正在研究Chebotarev密度定理的改进，以期应用高阶筛的数域版本。我的第三个博士生Francois Seguin正在研究与椭圆曲线相关的问题。**

In 1947, Atle Selberg discovered a new method in sieve theory which revolutionized the subject.  This method is now called the Selberg sieve and has been used in a spectrum of applications ranging from the classical twin prime problem to more sophisticated questions of counting points on algebraic varieties.  Recently, a special case of a "higher rank'' version of the Selberg sieve was applied by Maynard and Tao (independently) to improve and simplify upon Zhang's ground breaking work regarding infinitely many bounded gaps between consecutive prime numbers.  In joint work with my doctoral student, Akshaa Vatwani, I have developed a general higher rank version of the classical Selberg sieve.  The work of Maynard and Tao now appears as a special case of this more general sieve.  Clearly, there will be further applications of this new sieve method and we plan to apply it to an assortment of problems in the coming years.  It looks as if there will be some potential applications to the Artin primitive root problem.  In addition, one can formulate also a number field version of this higher rank sieve.  My student is already looking at this possibility in her doctoral thesis currently in progress, and has obtained some interesting results regarding bounded gaps between Gaussian primes.  That is, there is a fixed number B such that there are infinitely many Gaussian primes a+bi and c+di such that |a-c| and |b-d| are both bounded by B.  We also have a new proof of the Tauberian theorem and we expect more results in this setting.***Clearly, these recent discoveries represent a cusp in sieve theory.  Indeed, the relationship between the higher rank Selberg sieve and the classical Selberg sieve is similar to the discovery of muti-variable calculus and one-variable calculus.  For example, in the classical Selberg sieve, in an attempt to prove that there are infinitely many twin primes, the sieve method was applied to the single sequence n(n+2) instead of the two-tuple sequence (n, n+2).  It is historically interesting that Selberg had suggested the "higher rank'' approach to the sieve as far back as 1969, at the end of one of his papers, but clearly the idea went unnoticed until it was recently resurrected by Maynard and Tao.  Though one can study functions of several variables using a one-variable theory, there is a richer structure in the multi-variable theory.  Similar is the case with the classical sieve and the higher rank sieve. ***We also expect other applications of the higher rank Selberg sieve to the study of gaps between primes satisfying Chebotarev conditions especially when the base field is not the rational number field.  My doctoral student, Peng-Jie Wong, is now investigating Artin L-series and we are studying improvements to the Chebotarev density theorem, with a view to applying number field versions of the higher rank sieve.  Related questions, but in the setting of elliptic curves, are being studied by my third doctoral student, Francois Seguin.  **