Topics in analytic number theory and beyond

解析数论及其他主题

• 批准号: 36642-2011
• 负责人: Granville, Andrew
• 金额: $ 4.08万
• 依托单位: Université de Montréal
• 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=593026
• 关键词: Topics; analytic; number; theory; beyond

In 1859 Riemann gave an identity for the number of prime numbers up to x in terms of the zeros of the Riemann zeta-function (which is the analytic continuation of a function that is "naturally defined" only on half of the complex plane). This compelling identity has been the basis of the study of the distribution of prime numbers ever since and all textbooks start from this perspective. Indeed before the 1949 elementary proof of the prime number theorem it was believed to be impossible to take any different approach and even after that it had seemed that other approaches are "ad hoc", and so limited in their applicability.

1859年，黎曼用黎曼Zeta函数的零点给出了素数到x的个数的一个恒等式(Riemann Zeta函数是一个函数的解析延拓，该函数只在复平面的一半上“自然定义”)。从那时起，这个令人信服的身份一直是研究素数分布的基础，所有的教科书都是从这个角度出发的。事实上，在1949年素数定理的初等证明之前，人们认为不可能采取任何不同的方法，即使在那之后，似乎其他方法也是“临时的”，因此它们的适用性有限。