Averages of Artin characters and their relation to the Davenport--Heilbronn theorem
Abstract
We bound the error term in the counting function of cubic extensions of ℚ by 𝒪ε(Xθ+ε), with θ< 2/3, improving upon results due to Bhargava, Taniguchi, and Thorne. The proof proceeds by relating a certain discriminant-reducing sieve, due to Bhargava, Shankar, and Tsimerman, to averages of Artin characters associated with the Artin L-function ζK(s)/ζ(s), with K a cubic field. Conditional on the Generalised Riemann Hypothesis (GRH) for Dedekind zeta functions ζK(s), we obtain further savings and show that the error term is 𝒪(X2/3−1/51+ε). Moreover, for the analogous smooth counting problem, we obtain a bound 𝒪(X1/2+ε) for the error term, conditionally improving results of Shankar, Södergren, and Templier. Our arguments can also handle finitely many splitting conditions. The methods we use to study cubic fields are natural extensions of methods which are used to study the one-level density associated with the above family of Artin L-functions. In this direction, we improve the admissible support [−σ,σ], from the previously known admissible value σ< 2/5, to σ< 1, conditional on the GRH. This also proves that at least 75% of these L-functions are non-vanishing at the central point, conditionally improving upon results of Shankar, Södergren and Templier. Finally, one may study similar questions over a rational function field 𝔽q(T), with q coprime to $2$ and $3$. Here, we obtain a proportion of non-vanishing of at least 75% for the analogous family of L-functions. Furthermore, we obtain a bound 𝒪(X1/2+ε) for the error term in the counting function of cubic function fields, improving previous results of the author. Over function fields, our results are unconditional.
Related
- Orthogonality relation between GL(3) coefficients and additive characters with linear and non-linear phases
- A tensor square theorem for characters of GLn(q)
- Gallagher's Theorem for Real-valued Brauer Characters
- Heilbronn's Problem in the Unit Triangle: Computational Optimality Certificates for n ≤ 8
- KKT Stresses, Affine Moments, and Separator Flux in the Heilbronn Triangle Problem
BibTeX
@misc{ahlquist2026averages,
title = {{Averages of Artin characters and their relation to the Davenport--Heilbronn theorem}},
author = {Victor Ahlquist},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01837},
eprint = {2610.01837},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01837},
}
RIS
TY - UNPB
TI - Averages of Artin characters and their relation to the Davenport--Heilbronn theorem
AU - Victor Ahlquist
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.01837
UR - https://arxiv.org/abs/2610.01837
AB - We bound the error term in the counting function of cubic extensions of ℚ by 𝒪_ε(X^θ+ε), with θ< 2/3, improving upon results due to Bhargava, Taniguchi, and Thorne. The proof proceeds by relating a certain discriminant-reducing sieve, due to Bhargava, Shankar, and Tsimerman, to averages of Artin characters associated with the Artin L-function ζ_K(s)/ζ(s), with K a cubic field. Conditional on the Generalised Riemann Hypothesis (GRH) for Dedekind zeta functions ζ_K(s), we obtain further savings and show that the error term is 𝒪(X^2/3−1/51+ε). Moreover, for the analogous smooth counting problem, we obtain a bound 𝒪(X^1/2+ε) for the error term, conditionally improving results of Shankar, Södergren, and Templier. Our arguments can also handle finitely many splitting conditions. The methods we use to study cubic fields are natural extensions of methods which are used to study the one-level density associated with the above family of Artin L-functions. In this direction, we improve the admissible support [−σ,σ], from the previously known admissible value σ< 2/5, to σ< 1, conditional on the GRH. This also proves that at least 75% of these L-functions are non-vanishing at the central point, conditionally improving upon results of Shankar, Södergren and Templier. Finally, one may study similar questions over a rational function field 𝔽_q(T), with q coprime to $2$ and $3$. Here, we obtain a proportion of non-vanishing of at least 75% for the analogous family of L-functions. Furthermore, we obtain a bound 𝒪(X^1/2+ε) for the error term in the counting function of cubic function fields, improving previous results of the author. Over function fields, our results are unconditional.
ER -