The Mobius Function

DMS-0301168Iwaniec，HenrykAbstractTitle：Mobius函数该提议处理了用模形式的系数来逼近Mobius函数，特别是与正排序的椭圆曲线相关的系数。关键问题是实现相对于所采用的形式家族的基数的导体大小的高度一致性。基本工具是特征和。已经确定的估计数表明，有可能取得更好的结果。对于应用程序来说，即使是一个微小的改进也是至关重要的，特别是对于消除L函数的“例外零”。该提案包括几个猜想，这些猜想为解决主要问题奠定了基础。算术的基本目标之一是度量数域上唯一因式分解性质的失效。这一提议的动机是希望用解析的术语来解决这个经典问题，特别是通过给出所谓的类数的有效界限。这对于理解解析数论的现代工具也具有重要意义，并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.