Moments and non-vanishing of cubic Dirichlet L-functions at s =1/2
Heilbronn Number Theory Seminar
16th December 2020, 4:00 pm – 5:00 pm
A famous conjecture of Chowla predicts that L(1/2, \chi) is nonzero 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 =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.