The Mobius Function

DMS-0301168 Iwaniec，Henryk摘要标题：莫比乌斯函数 该提案处理近似的莫比乌斯函数的系数的模块形式，特别是这些与椭圆曲线的正秩。关键的问题是要达到高度的一致性，相对于正在使用的形式的家庭基数的导体的大小。 基本的工具是特征和指数和。已经确定的估计数表明，取得更好的结果是可能的。即使是一个微小的改进将是至关重要的应用程序，特别是消除“异常零”的L-函数。该提案包括几个说明，为解决主要问题指明了方向。 算术的基本目标之一是度量数域上唯一因子分解性质的失败。该提案的动机是希望解决这个经典问题的分析条款，特别是通过给一个有效的限制所谓的类数。 这对于理解解析数论的现代工具也将是重要的，并且可能吸引新的研究人员，因为有可能将这些方法应用于其他中心问题。

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.