The Higher Rank Selberg Sieve and Applications

1947年，Atle Selberg发现了一种新的筛理论方法，彻底改变了这个问题。 这种方法现在被称为塞尔伯格筛，并已被用于一系列的应用范围从经典的孪生素数问题，以更复杂的问题计数点的代数簇。 最近，梅纳德和陶（独立地）应用了塞尔伯格筛的“高阶”版本的特殊情况，以改进和简化张的开创性工作，即连续素数之间的无限多个有界间隙。 在与我的博士生Akshaa Vatwani的共同工作中，我开发了一个经典Selberg筛的一般高阶版本。 梅纳德和陶的工作现在看来是这个更一般的筛子的一个特例。 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。 看起来好像有一些潜在的应用程序的阿廷原根问题。 此外，人们还可以用公式表示这种高阶筛子的数域形式。 我的学生已经在她正在进行的博士论文中研究这种可能性，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，存在一个固定数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|都以B为界。 我们也有了陶伯定理的一个新证明，我们期待在这种情况下有更多的结果。显然，这些最近的发现代表了筛子理论的一个尖端。 事实上，高阶Selberg筛法和经典Selberg筛法之间的关系类似于多元微积分和一元微积分的发现。 例如，在经典的塞尔伯格筛法中，为了证明有无穷多个孪生素数，筛法被应用于单个序列n（n +2）而不是二元序列（n，n +2）。 有趣的是，早在1969年，塞尔伯格就在他的一篇论文的结尾提出了"更高等级"的筛子方法，但很明显，这个想法一直没有被注意到，直到最近梅纳德和陶重新提出。 虽然人们可以用一元理论来研究多元函数，但多元理论有更丰富的结构。 经典筛和高阶筛的情况类似。* 我们还期望高阶Selberg筛的其他应用，以研究满足Chebotarev条件的素数之间的间隙，特别是当基域不是有理数域时。 我的博士生Peng-Jie Wong现在正在研究Artin L-级数，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。 我的第三个博士生弗朗索瓦·塞甘正在研究相关的问题，但都是在椭圆曲线的背景下。 **

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.  **