Distributions of prime numbers and zeros of L-functions

解析数论的一个核心方面是研究素数在各种等差数列中的分布。这种数列的例子有等差数列、多项式的值、被常数移位的素数等等。亚纯函数，如黎曼ζ函数和狄利克雷l函数及其零点的位置对于理解素数的分布是至关重要的。在我的研究中，我试图量化一个有限等差数列比另一个包含更多素数的“概率”。我建议从理论和计算的角度进一步研究Rubinstein和Sarnak首先提出的这些概率。我目前正在开发一种理论，可以精确预测它们的大小，作为所考虑的等差数列的函数，我计划用概率的计算来测试这一理论。这些计算至少可以说是非平凡的，涉及到狄利克雷l函数的零点列表以及精细的数值积分。我还计划解决高维概率的类似问题，例如，同时涉及三个等差数列。假设黎曼ζ函数或狄利克雷l函数的零点构型可以暗示素数分布的不规则性。我建议遵循Ford和Konyagin的想法来确定这些配置如何影响上述概率。此外，Green和Tao最近关于质数的长等差数列的结果预示着解析数论方法的突破。我计划研究他们的方法的可能扩展，特别是如何包含狄利克雷l函数的零信息来处理多项式的素值，例如。

One of the central aspects of analytic number theory is studying the distribution of prime numbers in various sequences of arithmetic interest. Examples of such sequences are arithmetic progressions, values of polynomials, primes shifted by a constant, and so on. Meromorphic functions such as the Riemann zeta function and Dirichlet L-functions and the locations of their zeros are of paramount importance to the understanding of the distribution of primes. In my research, I attempt to quantify the "probability" that one finite arithmetic progression contains more primes than another. I propose to further study these probabilities, first put forward by Rubinstein and Sarnak, from both theoretical and computational viewpoints. I am currently developing a theory that makes precise predictions of their sizes as functions of the arithmetic progressions under consideration, and I plan to test this theory against computations of the probabilities. These computations are nontrivial to say the least, involving lists of zeros of Dirichlet L-functions as well as delicate numerical integrations. I also plan to address the analogous questions for higher-dimensional probabilities that involve, for example, three arithmetic progressions simultaneously. Hypothetical configurations of zeros of the Riemann zeta function or Dirichlet L-functions can imply irregularities in the distribution of prime numbers. I propose to follow the ideas of Ford and Konyagin in determining how such configurations could affect the probabilities described above. Moreover, the recent result of Green and Tao on long arithmetic progressions of primes heralds a breakthrough in methods of analytic number theory. I plan to study possible extensions to their method, in particular how information on zeros of Dirichlet L-functions can be included to address prime values of polynomials, for example.