Graduate Number Theory Seminar

Title: The \theta=\infty Conjecture and the Riemann Hypothesis

Abstract: The Riemann Hypothesis is closely connected to the behavior of moments of the Riemann zeta function. In particular, a conjecture of Bettin and Gonek asserts that suitably normalized mollified second moments remain bounded even when the mollifier has arbitrarily large polynomial length. This is known as the \theta=\infty conjecture.

In this talk, we will discuss an extension of this conjecture to automorphic L-functions and its consequences for the distribution of their zeros. We will see how bounds for mollified moments imply zero-free regions and how an averaged version of the conjecture leads to a quasi-Riemann Hypothesis for families of automorphic L-functions.