The Mobius Function

莫比乌斯函数

• 批准号: 0301168
• 负责人: Henryk Iwaniec
• 金额: $ 57.55万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0301168&HistoricalAwards=false
• 关键词: Mobius; Function

DMS-0301168Iwaniec, HenrykAbstractTitle: The Mobius Function The Proposal deals with approximations to the Mobius function by coefficients of modular forms, particularly of these associated with elliptic curves of positive rank. The key problem is to achieve high uniformity with respect to the size of the conductor relatively to the cardinality of the family of forms being employed. The basic tools are character sums and exponential sums. The estimates already established indicate that stronger results are possible. Even a slight improvement will be critical for applications, especially for the elimination of the "exceptional zero" of L-functions. The Proposal includes several conjectures which set the way to attack the main problem. One of the fundamental goals of arithmetic is to measure the failure of unique factorization property in number fields. The Proposal is motivated by a desire to solve this classical problem in analytic terms, specifically by giving an effective bound for the so called class number. This will be significant also for understanding the modern tools of analytic number theory, and may attract new researchers because of possibilities to apply these methods to other central issues.

题目：莫比乌斯函数本文讨论了用模形式的系数逼近莫比乌斯函数，特别是与正秩椭圆曲线相关的模形式的系数逼近。关键问题是实现导体尺寸相对于所采用的形式家族的基数的高度均匀性。基本的工具是特征和和和指数和。已经确定的估计表明，有可能取得更大的成果。即使是一个微小的改进对于应用程序来说也是至关重要的，特别是对于消除l函数的“异常零”。该建议包括几个猜想，为解决主要问题开辟了道路。算法的基本目标之一是测量数字域的唯一分解性质是否失效。这个提议的动机是希望用解析的方式解决这个经典问题，特别是通过给出所谓的类数的有效界。这对于理解解析数论的现代工具也很重要，并可能吸引新的研究人员，因为这些方法有可能应用于其他中心问题。