Moments and non-vanishing of cubic Dirichlet L-functions at s=1/2

Abstract: A famous conjecture of Chowla predicts that $L(\frac{1}{2},\chi)\neq 0$ for all Dirichlet L-functions attached to primitive characters $\chi$. It was conjectured first in the case where $\chi$ is a quadratic character, which is the most studied case. For quadratic Dirichlet L-functions, Soundararajan proved that at least 87.5% of the quadratic Dirichlet L-functions do not vanish at $s=\frac{1}{2}.$

Under GRH, there are slightly stronger results by Ozlek and Snyder. We present in this talk the first result showing a positive proportion of cubic Dirichlet L-functions non-vanishing at s = 1/2 for the non-Kummer case over function fields. This can be achieved by using the recent breakthrough work on sharp upper bounds for moments of Soundararajan, Harper and Lester-Radziwill. Our results would transfer over number fields (but we would need to assume GRH in this case). The talk will be accessible to a general audience of number theorists and graduate students in number theory.

Joint work with A. Florea and M. Lalin.

number theory

Audience: researchers in the discipline

Upstate New York Online Number Theory Colloquium