发表机构
Chalmers University of Technology; University of Gothenburg（查尔姆斯理工大学; 哥德堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过Artin特征平均值改进了三次扩张计数函数的误差项界，在GRH下达到$X^{2/3-1/51+\epsilon}$，并证明至少75%的L-函数在中心点非零，且函数域结果无条件。
AI 中文摘要
我们将$\mathbb{Q}$的三次扩张计数函数中的误差项界定为$\mathcal{O}_\epsilon\left(X^{\theta+\epsilon}\right)$，其中$\theta < 2/3$，改进了Bhargava、Taniguchi和Thorne的结果。证明过程通过将Bhargava、Shankar和Tsimerman提出的某个判别式缩减筛法与Artin $L$-函数$\zeta_K(s)/\zeta(s)$（其中$K$为三次域）相关的Artin特征平均值联系起来。在Dedekind zeta函数$\zeta_K(s)$的广义黎曼假设（GRH）条件下，我们获得进一步的节省，并证明误差项为$\mathcal{O}\left(X^{2/3-1/51+\epsilon}\right)$。此外，对于类似的平滑计数问题，我们在条件性改进了Shankar、Södergren和Templier的结果后，获得误差项的界$\mathcal{O}(X^{1/2+\epsilon})$。我们的论证还可以处理有限多个分裂条件。我们用于研究三次域的方法是研究上述Artin $L$-函数族相关的一级密度所用方法的自然扩展。在这一方向上，我们在GRH条件下将可容许支撑$[-\sigma,\sigma]$从先前已知的可容许值$\sigma < 2/5$改进到$\sigma < 1$。这也证明了这些$L$-函数中至少有$75\\%$在中心点非零，条件性改进了Shankar、Södergren和Templier的结果。最后，我们可以在有理函数域$\mathbb{F}_q(T)$上研究类似问题，其中$q$与$2$和$3$互素。在此，我们为类似的$L$-函数族获得至少$75\\%$的非消失比例。此外，我们为三次函数域的计数函数中的误差项获得界$\mathcal{O}(X^{1/2+\epsilon})$，改进了作者先前的结果。在函数域上，我们的结果是无条件的。
英文摘要
We bound the error term in the counting function of cubic extensions of $\mathbb{Q}$ by $\mathcal{O}_ε\left(X^{θ+ε}\right)$, 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 $\mathcal{O}\left(X^{2/3-1/51+ε}\right)$. Moreover, for the analogous smooth counting problem, we obtain a bound $\mathcal{O}(X^{1/2+ε})$ 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 $\mathbb{F}_q(T)$, 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 $\mathcal{O}(X^{1/2+ε})$ 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.
Comments42 pages
