Omega results for cubic field counts via lower-order terms in the one-level density

TL;DR

This paper derives a precise 1-level density formula for non-Galois cubic L-functions, revealing a new lower-order term that leads to an omega result for cubic field counts and refines the Ratios Conjecture.

Contribution

It introduces a novel secondary term in the 1-level density for cubic L-functions, impacting predictions of the Ratios Conjecture and cubic field counting error bounds.

Findings

Discovery of a unique secondary term in the 1-level density.

Derivation of an omega result for cubic field counting functions.

Refinement of the Ratios Conjecture to include the new lower-order term.

Abstract

In this paper we obtain a precise formula for the $1$-level density of $L$-functions attached to non-Galois cubic Dedekind zeta functions. We find a secondary term which is unique to this context, in the sense that no lower-order term of this shape has appeared in previously studied families. The presence of this new term allows us to deduce an omega result for cubic field counting functions, under the assumption of the Generalized Riemann Hypothesis. We also investigate the associated $L$-functions Ratios Conjecture, and find that it does not predict this new lower-order term. Taking into account the secondary term in Roberts' Conjecture, we refine the Ratios Conjecture to one which captures this new term. Finally, we show that any improvement in the exponent of the error term of the recent Bhargava--Taniguchi--Thorne cubic field counting estimate would imply that the best possible error term in the refined Ratios Conjecture is $O_\varepsilon(X^{-\frac 13+\varepsilon})$. This is in opposition with all previously studied families, in which the expected error in the Ratios Conjecture prediction for the $1$-level density is $O_\varepsilon(X^{-\frac 12+\varepsilon})$.