Non-vanishing of primitive cubic, quartic and sextic L-functions
Abstract
We establish lower bounds for the non-vanishing proportions of primitive cubic, quartic and sextic Dirichlet L-functions over Fq[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−e53 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.
Related
- Improved first moments of primitive quartic and sextic Dirichlet L-functions
- Large values of cubic and quartic Dirichlet L-functions of prime moduli
- A secondary term in the first moment of primitive cubic Dirichlet L-functions
- A Sarkisov Involution from a Rational Cubic Curve on a Quartic Double Solid
- Benign cubic and quartic forms on the sphere
BibTeX
@misc{hong2026nonvanishing,
title = {{Non-vanishing of primitive cubic, quartic and sextic L-functions}},
author = {Ziwei Hong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15316},
eprint = {2610.15316},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15316},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Non-vanishing of primitive cubic, quartic and sextic L-functions
AU - Ziwei Hong
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.15316
UR - https://arxiv.org/abs/2610.15316
AB - 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.
ER -