Moments of primes in arithmetic progressions

算术级数中素数的矩

• 批准号: RGPIN-2015-05955
• 负责人: Fiorilli, Daniel
• 金额: $ 1.38万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=613848
• 关键词: Moments; primes; arithmetic; progressions

My research project is concerned with the distribution of primes in arithmetic progressions, that is primes of the form p=qn+a, with a and q two coprime integers which are determined in advance. The distribution of prime numbers is a central question in number theory, and has many applications. Perhaps the most omnipresent practical application of prime numbers is the RSA cryptography algorithm, which is used every time a credit card transaction is being processed on the internet. See for example the nice Slate article http://www.slate.com/articles/health_and_science/science/2013/06/online_credit_card_security_the_rsa_algorithm_prime_numbers_and_pierre_fermat.html. On a more theoretical note, the understanding of primes in arithmetic progressions constitutes a fundamental step in many other number theoretical problems, such as for example the revolutionary theorems of Zhang, Maynard and Tao which imply that there are infinitely many pairs of primes which differ by at most 246.  My main goal is to understand several statistics (chiefly moments) of the distribution of primes p=qn+a up to a given limit x. The knowledge of all moments is usually sufficient to uniquely determine a distribution. However, the first two moments, the mean and variance, already give a good understanding of a distribution, and imply certain "almost everywhere" statements. These two moments have been extensively studied in the literature by many well-known mathematicians. For instance, Hooley devoted more than fifteen research papers on questions related to the variance. For the first moment, I plan to extend the results of my thesis to different arithmetical contexts, and to introduce Vaughan's approximation in order to obtain more precise results. The range where this last approximation works best is precisely the range where my thesis results apply, and it is clear that sharp results will follow from combining the techniques I used with this approximation. Vaughan introduced his approximation in a paper where he studied the variance of primes in arithmetic progressions. I plan to revise his analysis and to hopefully sharpen his results, by using an alternative technique. As for higher moments, I plan to apply probabilistic arguments using the explicit formula, as was already done with the variance in my previous work.  The different techniques I will use are quite complementary in that they apply to very different ranges. Probabilistic arguments on explicit formulas work for very small moduli, zero-statistic arguments with the help of random matrix theory work for intermediate moduli, and divisor-switching techniques work for large moduli. The combination of all these techniques should allow for a better understanding of the distribution of primes in arithmetic progressions.

我的研究项目涉及算术级数中素数的分布，即 p=qn+a 形式的素数，其中 a 和 q 是预先确定的两个互质整数。素数的分布是数的中心问题。质数最普遍的实际应用可能是 RSA 加密算法，每次在互联网上处理信用卡交易时都会使用该算法，例如，请参阅 Slate 文章。 http://www.slate.com/articles/health_and_science/science/2013/06/online_credit_card_security_the_rsa_algorithm_prime_numbers_and_pierre_fermat.html 从更理论的角度来看，对算术级数中素数的理解构成了许多其他数论问题的基本步骤，例如例如，张、梅纳德和陶的革命定理意味着有无限多对素数，最多相差 246。 我的主要目标是了解素数 p=qn+a 到给定极限 x 的分布的几个统计数据（主要是矩）。所有矩的知识通常足以唯一地确定一个分布。前两个矩，即均值和方差，已经很好地理解了分布，并且暗示了某些“几乎无处不在”的陈述，这两个时刻已被许多著名数学家在文献中进行了主要研究，例如，胡利在这方面投入了超过十五篇研究论文。与方差有关的问题。首先，我计划将我的论文结果扩展到不同的算术环境，并引入沃恩近似，以获得更精确的结果，最后一个近似效果最好的范围正是我的论文结果适用的范围，显然，将我使用的技术与这种近似相结合，沃恩在一篇研究算术中素数方差的论文中介绍了他的近似值，将会得出清晰的结果。我计划修改他的分析，并希望通过使用替代技术来提高他的结果，对于更高的时刻，我计划使用显式公式应用概率论证，就像我之前的工作中已经对方差所做的那样。 我将使用的不同技术是相当互补的，因为它们适用于非常不同的范围，显式公式的概率论证适用于非常小的模数，借助随机矩阵理论适用于中间模数，以及除数切换技术。所有这些技术的结合应该可以更好地理解算术级数中素数的分布。