Moments of L-functions

This page is for the keynote presentation by Dr. Aliah Hamieh.

Abstract

An L-function \( L(s,X) \) is a meromorphic function on the complex plane that is usually associated with an object \( X \) of arithmetic, algebraic or geometric nature. A central theme in analytic number theory is to understand the behavior of L-functions inside the critical strip \( 0 < \text{Re}(s) < 1 \). The generalized Lindelöf Hypothesis is a statement about the size of L-functions in the critical strip, and it motivates the study of moments of L-functions as a crucial tool for investigation. In fact, establishing asymptotic formulae for moments of L-functions has become a focal point in analytic number since the early 1900’s. A major breakthrough in this area occurred in 1998 when Keating and Snaith established a conjectural formula for moments of the Riemann zeta function using ideas from random matrix theory. The methods of Keating and Snaith led to similar conjectures for moments of many families of L-functions. These conjectures have become a driving force in this field which has witnessed substantial progress in the last two decades.

In this talk, I will review the history of this subject and survey some recent results. If time permits, I will discuss recent joint work with Nathan Ng and Fatma Cicek on the mean values of long Dirichlet polynomials which could be used to model moments of the zeta function.