L-functions and Multiplicative Number Theory

调查员和他的同事研究各种问题的解析理论的L-函数和平均值的multiplicativefunctions。 特别是调查的目的是提高理解的零点分布的L-函数，他们的行为在特殊值，和极端值的大小在临界地带。 本项目的第二个主要目的是描述乘法函数的可能平均值的谱，这些函数在素数处的值被限制在单位圆的特定子集中。 研究者与Granville合作，打算进一步发展这种平均值与一族积分方程的解之间的密切联系。 在这一领域的一个基本问题是理解黎曼zeta函数和其他L-函数的零点。 这些函数编码了很多关于素数和其他算术对象的信息。 除了数学家的内在兴趣之外，从这些问题中产生的想法已经在计算机科学，密码学和编码理论中找到了用途。

The investigator and his colleagues study various problems inthe analytic theory of L-functions and mean values of multiplicativefunctions. In particular the investigator aims for an improvedunderstanding of the distribution of zeros of L-functions,their behaviour at special values, and the size of extreme valuesin the critical strip. The second main aim of this project isto describe the spectrum of possible mean-values of multiplicativefunctions whose values at primes are constrained to lie in specialsubsets of the unit disc. The investigator, in collaborationwith Granville, intends to develop further the close connection betweensuch mean values and solutions to a family of integral equations.This project concerns problems in the area of multiplicativenumber theory. A fundamental problem in this area is to understandthe zeros of the Riemann zeta function and other L-functions. Thesefunctions encode much information about prime numbers, andother arithmetic objects. Besides being of intrinsic interest tomathematicians, ideas emerging from these problems have found usein computer science, cryptography and coding theory.