The Higher Rank Selberg Sieve and Applications

1947年，Atle Selberg在筛子理论中发现了一种新的方法，它彻底改变了这一主题。这种方法现在被称为Selberg筛子，并已被用于从经典的孪生素数问题到更复杂的代数簇上的计点问题的一系列应用中。最近，Maynard和Tao(独立)应用了Selberg筛子的一个特例，以改进和简化张关于连续素数之间无穷多个有界间隙的开创性工作。我与我的博士生Akshaa Vatwani共同工作，我已经开发了经典Selberg筛子的一个一般的高阶版本。而Maynard和陶的工作现在似乎是这个更一般筛子的一个特例。显然，这种新的筛子方法还会有更多的应用，我们计划在未来几年将它应用于各种问题。看起来似乎会有一些潜在的应用于Artin原根问题。此外，人们还可以制定这个高阶筛子的数域版本。我的学生已经在她目前正在进行的博士论文中研究这种可能性，并得到了一些关于高斯素数之间的有界的有界结果。也就是说，有一个固定的数B，使得存在无穷多个高斯素数a+bi和c+di，使得|a-c|和|b-d|都有B有界。我们也有了陶伯利亚定理的一个新证明，我们希望在这个背景下有更多的结果。*显然，这些最近的发现代表着筛子理论的一个尖端。事实上，高阶Selberg筛子和经典Selberg筛子之间的关系类似于多元微积分和单变量微积分的发现。例如，在经典的Selberg筛子中，试图证明存在无穷多个孪生素数，筛法被应用于单序列n(n+2)而不是两元组序列(n，n+2)。历史上很有趣的是，Selberg早在1969年就在他的一篇论文的末尾提出了筛法，但显然这个想法没有被注意到，直到最近Maynard和Tao复活了它。尽管一个人可以用单变量理论来研究多个变量的函数，但在多变量理论中有一个更丰富的结构。经典筛子和高阶筛子的情况类似。*我们也期待高阶Selberg筛在研究满足切博塔雷夫条件的素数之间的间隔方面的其他应用，特别是当基域不是有理数域时。我的博士生王鹏杰现在正在研究Artin L级数，我们正在研究对切博塔夫密度定理的改进，以期应用高阶筛的数域版本。我的第三个博士生Francois Sguin正在研究其他相关的问题，但在椭圆曲线的设置下。

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.  **