Moments of primes in arithmetic progressions

我的研究项目是关于等差数列中素数的分布，即p=qn+a形式的素数，其中a和q两个素数整数是预先确定的。素数的分布是数论中的一个核心问题，有着广泛的应用。也许素数最普遍的实际应用是RSA加密算法，每次在互联网上处理信用卡交易时都会用到它。参见nice 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。从更理论化的角度来看，对等差数列中素数的理解是解决许多其他数论问题的一个基本步骤，例如张、梅纳德和陶的革命性定理，该定理表明存在无限多对素数，它们的差不超过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，已经让我们很好地理解了一个分布，并暗示了某些“几乎无处不在”的陈述。这两个时刻被许多著名的数学家广泛地研究过。例如，胡利在与方差相关的问题上发表了超过15篇研究论文。首先，我打算将我的论文结果扩展到不同的算术环境中，并引入Vaughan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，很明显，将我使用的技术与这个近似相结合，将会产生清晰的结果。沃恩在一篇研究等差数列中质数方差的论文中介绍了他的近似。我打算修改他的分析，并希望通过使用另一种技术来提高他的结果。对于更高的矩，我打算使用显式公式应用概率参数，就像我之前的工作中对方差所做的那样。我将使用的不同技术是互补的，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，随机矩阵理论帮助下的零统计论证适用于中间模，而因子交换技术适用于大模。所有这些技术的结合可以让我们更好地理解等差数列中素数的分布

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.**