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 $\mathbb{Q}$ by $\mathcal{O}_\epsilon\left(X^{\theta+\epsilon}\right)$, with $\theta<2/3$, 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 $\zeta_K(s)/\zeta(s)$, with $K$ a cubic field. Conditional on the Generalised Riemann Hypothesis (GRH) for Dedekind zeta functions $\zeta_K(s)$, we obtain further savings and show that the error term is $\mathcal{O}\left(X^{2/3-1/51+\epsilon}\right)$. Moreover, for the analogous smooth counting problem, we obtain a bound $\mathcal{O}(X^{1/2+\epsilon})$ for the error term, conditionally improving results of Shankar, S\"odergren, 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 $[-\sigma,\sigma]$, from the previously known admissible value $\sigma<2/5$, to $\sigma<1$, 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, S\"odergren 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 $\mathcal{O}(X^{1/2+\epsilon})$ 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.