Moments of primes in arithmetic progressions

我的研究项目是关于素数在算术级数中的分布，即形式为p=qn+a的素数，其中a和q是预先确定的两个互素整数。素数的分布是数论中的一个中心问题，有很多应用。素数最无处不在的实际应用可能是RSA密码算法，每次在互联网上处理信用卡交易时都会使用该算法。在一个更理论性的笔记上，对算术级数中素数的理解构成了许多其他理论问题的基本步骤，例如张、梅纳德和陶渊明的革命性定理，这些定理暗示着有无限多对素数至多相差246%。http://www.slate.com/articles/health_and_science/science/2013/06/online_credit_card_security_the_rsa_algorithm_prime_numbers_and_pierre_fermat.html. 我的主要目标是了解素数p=qn+a到给定极限x的分布的几个统计量(主要是矩)。所有矩的知识通常足以唯一地确定一个分布。然而，前两个时刻，均值和方差，已经很好地理解了分布，并暗示了某些“几乎无处不在”的陈述。许多著名数学家在文献中对这两个时刻进行了广泛的研究。例如，胡利花了15篇以上的研究论文来研究与差异有关的问题。首先，我计划将我的论文的结果推广到不同的算术环境中，并引入沃恩的近似，以获得更精确的结果。最后一个近似最有效的范围正好是我的论文结果适用的范围，很明显，将我使用的技术与这个近似结合起来，将会得出明显的结果。沃恩在一篇研究算术级数中素数的方差的论文中介绍了他的近似。我计划修改他的分析，并希望通过使用另一种技术来提高他的结果。至于更高的时刻，我计划使用显式的概率论点，就像我在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。显式公式上的概率论证适用于非常小的模数，零统计论证借助随机矩阵理论适用于中间模数，除数转换技术适用于大模数。所有这些技术的结合应该可以更好地理解素数在算术级数中的分布。

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.