K. Soundararajan (University of Michigan)

What are L-functions and what are they good for?
L-functions are analytic objects which encode arithmetical information such as prime numbers, class numbers of fields, the number of rational points on elliptic curves etc. The prototypical example of an L-function is Riemann's zeta function. Understanding the behavior of L-functions leads naturally to an understanding of many number theoretic questions. I will give many examples of L-functions, and describe the central problems of this theory. I will also give several applications of L-functions to concrete problems in number theory.

The mollifier method and non-vanishing results for L-functions
The mollifier method originates in Selberg's celebrated result that a positive proportion of zeros of the Riemann zeta function lie on the critical line. The term was coined by Levinson in his proof that this proportion is at least a third. Nowadays the mollifier method has been used very successfully to establish non-vanishing results for L-functions at special values. I will give a description of the method together with some applications.

The distribution of values of L-functions
In this lecture I will discuss several questions regarding the value distribution of L-functions. In particular I will describe how L-functions can be modelled very well at the edge of the critical strip, and how this (presumably) extends to points not on the critical line. The situation on the critical line is quite different and very poorly understood, and I will describe some of the recent speculations/conjectures on this along with some partial progress.