Chowla conjectured that L(1/2,๐) never vanishes, for ๐ any Dirichlet character. Soundararajan showed that more than 87.5% of the values L(1/2,๐d), for ๐d a quadratic character, do not vanish. Much less is known about cubic characters. Baier and Young showed that more than X6/7-๐บ of L(1/2,๐) are non-vanishing, for ๐ a primitive, cubic character of conductor of size up to X. I will talk about recent joint work with C. David and M. Lalin, where we show that a positive proportion of these central L-values are non-vanishing in the function field setting. This is achieved by computing the first mollified moment using techniques previously developed by the authors in their work on the first moment of cubic Lโfunctions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Soundararajan, Harper and LesterโRadziwill.
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
