The Mobius Function

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

DMS-0301168 Iwaniec, Henryk Abstract Title: 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.