Non-vanishing for cubic $L$--functions

Abstract

We prove that there is a positive proportion of $L$-functions associated to cubic characters over ${\mathbb{F}}_q[T]$ that do not vanish at the critical point $s=1/2$. 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 $L$-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester-Radziwill, which in turn develops further ideas from the work of Soundararajan, Harper, and Radziwill-Soundararajan. We work in the non-Kummer setting when $q\equiv 2(\mathrm{mod}3)$, but our results could be translated into the Kummer setting when $q\equiv 1(\mathrm{mod}3)$ as well as into the number field case (assuming the Generalized Riemann Hypothesis). Our positive proportion of non-vanishing is explicit, but extremely small, due to the fact that the implied constant in the upper bound for the mollified second moment is very large.

53 pages

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