Year: 2026
Type: preprint
Abstract: We establish lower bounds for the non-vanishing proportions of primitive cubic, quartic and sextic Dirichlet $L$-functions over $\F_q[T]$ in the non-Kummer setting. Adapting truncated-exponential techniques for shifted moments, we improve the cubic lower bound of David, Florea and Lalín and obtain positive lower bounds for quartic and sextic families. For each family, the lower limit of the non-vanishing proportion exceeds $e^{-e^{53}}$ as the conductor degree tends to infinity with $q$ fixed. We use a common choice of truncation parameters and explicitly control the exceptional sets. The cubic argument uses the published twisted first moment. We also establish uniform twisted first moments in the quartic and sextic cases, yielding an asymptotic formula for the sextic first moment.
Source: arXiv (Cornell University)
Author: Hong, Ziwei
Institutions: UNSW Sydney, Renmin University of China
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221834161