Distributions of prime numbers and zeros of L-functions

L 函数的素数和零点的分布

• 批准号: 250190-2006
• 负责人: Martin, Greg
• 金额: $ 1.17万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=363664
• 关键词: Distributions; prime; numbers; zeros; functions

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.

解析数论的一个中心方面是研究素数在各种算术序列中的分布。这样的序列的例子是算术级数，多项式的值，素数移一个常数，等等。亚纯函数，如黎曼zeta函数和狄利克雷L-函数及其零点的位置是至关重要的了解素数的分布。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"。我建议进一步研究这些概率，首先提出的鲁宾斯坦和Sarnak，从理论和计算的角度。我目前正在开发一种理论，可以精确预测它们的大小，作为所考虑的算术级数的函数，我计划用概率计算来测试这个理论。这些计算至少可以说是不平凡的，涉及Dirichlet 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，例如同时涉及三个算术级数。黎曼zeta函数或狄利克雷L-函数的零点的假设配置可以暗示素数分布的不规则性。我建议遵循福特和科尼亚金的想法，以确定这种配置如何影响上述概率。此外，最近的结果绿色和陶长算术级数的素数预示着突破的方法分析数论。我计划研究他们的方法可能的扩展，特别是如何将Dirichlet L-函数的零点信息包括在内，以解决多项式的素数值。