Moments and non-vanishing of cubic Dirichlet $L$-functions at $s =\frac{1}{2}$

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 $s =\frac{1}{2}$ 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.

number theory

Audience: researchers in the topic

Heilbronn number theory seminar

Series comments: This is part of the University of Bristol's Heilbronn number theory seminar. If you wish to attend the talk (and are not a Bristolian), please register using this form or email us at bristolhnts@gmail.com with your name and affiliation (if any).

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