The Higher Rank Selberg Sieve and Applications

高阶塞尔伯格筛及其应用

• 批准号: RGPIN-2015-03957
• 负责人: Murty, Ram
• 金额: $ 2.99万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=688381
• 关键词: Higher; Rank; Selberg; Sieve; Applications

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.  **

1947年，Atle Selberg在筛法中发现了一种新的方法，彻底改变了筛法。这种方法现在被称为Selberg筛法，并已被用于从经典的孪生素数问题到代数簇上计数点的更复杂问题的一系列应用中。最近，梅纳德和陶应用了塞尔伯格筛的“高阶”版本的一个特例（独立）为了改进和简化张的开创性工作，关于连续素数之间的无限多个有界间隔。在与我的博士生的联合工作中，Akshaa Vatwani，我已经开发了一个更高级别的经典Selberg筛的版本。Maynard和Tao的工作现在作为这个更一般的筛的特例出现。显然，这种新的筛法将有进一步的应用，我们计划在未来几年内将其应用于各种各样的问题，看起来似乎对Artin原根有一些潜在的应用问题。此外，人们还可以用公式表示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，存在一个固定的数B，使得存在无限多个高斯素数a+bi和c+di，使得|a-C|和|B-D|都被B限制。我们也有了陶伯定理的新证明，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风口浪尖。事实上，高阶塞尔伯格筛子和经典塞尔伯格筛子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，在经典塞尔伯格筛子中，为了证明有无限多个孪生素数，筛选方法被应用于单序列n（n+2）而不是二元序列（n，n+2）。历史上有趣的是，早在1969年，Selberg在他的一篇论文的结尾就提出了筛选的“高阶”方法，但很明显，这个想法一直被忽视，直到最近由Maynard和Tao复活。虽然人们可以用一元理论研究多个变量的函数，但在多元理论中有更丰富的结构。类似的情况与经典筛和高阶筛。* 我们也期待高阶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。相关问题，但在椭圆曲线的设置中，我的第三个博士生，弗朗索瓦塞甘正在研究。