A secondary term in the first moment of cubic L-functions over function fields
Abstract
Let q be a fixed odd prime power with q≡2pmod3. We study the first moment of primitive cubic Dirichlet L-functions over Fq(T), ordered by their genus. By averaging the rational sieve variable in the double Dirichlet series of Hong and Fang, we obtain meromorphic continuation beyond the secondary polar divisor. Combining this continuation with the cubic Gauss-sum residues computed by David, Florea and Lalín gives an explicit term of order q5g/6, with a real coefficient depending on g modulo $6$, and an error Oq,eps(q(23/28+eps)g). The periodic coefficient is not identically zero. The proof uses an exact first difference supplied by the functional equation, an Euler product for the sieve average, and Bochner's tube theorem in the Dirichlet variables.
Related
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BibTeX
@misc{hong2026secondary,
title = {{A secondary term in the first moment of cubic $L$-functions over function fields}},
author = {Ziwei Hong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33505},
eprint = {2610.33505},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33505},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A secondary term in the first moment of cubic L-functions over function fields
AU - Ziwei Hong
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33505
UR - https://arxiv.org/abs/2610.33505
AB - Let q be a fixed odd prime power with q≡2pmod3. We study the first moment of primitive cubic Dirichlet L-functions over Fq(T), ordered by their genus. By averaging the rational sieve variable in the double Dirichlet series of Hong and Fang, we obtain meromorphic continuation beyond the secondary polar divisor. Combining this continuation with the cubic Gauss-sum residues computed by David, Florea and Lalín gives an explicit term of order q^5g/6, with a real coefficient depending on g modulo $6$, and an error O_q,eps(q^(23/28+eps)g). The periodic coefficient is not identically zero. The proof uses an exact first difference supplied by the functional equation, an Euler product for the sieve average, and Bochner's tube theorem in the Dirichlet variables.
ER -